自然哲学的数学原理-24

Imagine several concentric similarspheres, AB, CD, EF, &c.. the innermost of which added to the outermostmay compose a matter more dense towards the centre, or subducted fromthem may leave the same more lax andrare. Then, by Prop. LXXV, thesesphere? will attract other similar conSEC. XII.] OF NATURAL PHILOSOPHY. 223eentric spheres GH; IK, LM, &c., each the other, with forces reciprocallyproportional to the square of the distance SP. And, by composition ordivision, the sum of all those forces, or the excess of any of them abovethe others; that is, the entire force with which the whole sphere AB (composed of any concentric spheres or of their differences) will attract thewhole sphere GH (composed of any concentric spheres or their differences)in the same ratio. Let the number of the concentric spheres be increasedin infinitum, so that the density of the matter together with the attractiveforce may, in the progress from the circumference to the centre, increase ordecrease according to any given law ; and by the addition of matter not attractive, let the deficient density be supplied, that so the spheres may acquireany form desired ; and the force with which one of these attracts the otherwill be still, by the former reasoning, in the same ratio of the square of thedistance inversely. Q.E.I).COR. I. Hence if many spheres of this kind, similar in all respects, attract each other mutually, the accelerative attractions of each to each, atany equal distances of the centres, will be as the attracting spheres.COR. 2. And at any unequal distances, as the attracting spheres appliedto the squares of the distances between the centres./ COR. 3. The motive attractions, or the weights of the spheres towardsone another, will be at equal distances of the centres as the attracting andattracted spheres conjunctly ; that is, as the products arising from multiplying the spheres into each other.COR. 4. And at unequal distances, as those products directly, and thesquares of the distances between the centres inversely.COR. 5. These proportions take place also when the attraction arisesfrom the attractive virtue of both spheres mutually exerted upon eachother. 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 abouteach ; and the distances between the centres of the quiescent and revolvingbodies are proportional to the diameters of the quiescent bodies ; the periodic times will be equal.COR. 7. And, again, if the periodic times are equal, the distances willbe proportional to the diameters.COR. 8. All those truths above demonstrated, relating to the motionsjf bodies about the foci of conic sections, will take place when an attracting sphere, of any form and condition like that above described, is placedin the focus.COR. 9. And also when the revolving bodies are also attracting spheresDf any condition like that above described.224 THE MATHEMATICAL PRINCIPLES [BOOK I.PROPOSITION LXXVI1. THEOREM XXXVII.Tf to 1he several points of spheres there tend centripetal forces proportional to the distances of the points from the attracted bodies ; I say,that the compounded force with which two spheres attract each othermutually is as the distance between the centres of the spheres.CASE 1. Let AEBF be a sphere ; S itscentre . P a corpuscle attracted : PASBthe axis of the sphere passing through thecentre of the corpuscle ; EF, ef two planescutting the sphere, and perpendicular tothe axis, and equi-distant, one on one side,the other on the other, from the centre ofthe sphere ; G and g- the intersections ofthe planes and the axis ; and H any point in the plane EF. The centripetal force of the point H upon the corpuscle P, exerted in the direction ofthe line PH, is as the distance PH ; and (by Cor. 2, of the Laws) the sameexerted in the direction of the line PG, or towards the . centre S, is as thelength 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 thecentre 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 isattracted towards the centre S, is as that plane drawn into its distance Pg,or as the equal plane EF drawn into that distance Pg* ; and the sum of theforces of both planes as the plane EF drawn into the sum of the distancesPG + P^, that is, as that plane drawn into twice the distance PS of thecentre and the corpuscle ;that is, as twice the plane EF drawn into the distance PS, or as the sum of the equal planes EF + ef drawn into the samedistance. And, by a like reasoning, the forces of all the planes in thewhole sphere, equi-distant on each side from the centre of the sphere, areas the sum of those planes drawn into the distance PS, that is, as thewhole sphere and the distance PS conjunctly. Q,.E.D.CASE 2. Let now the corpuscle P attract the sphere AEBF. And, bythe same reasoning, it will appear that the force with which the sphere isattracted 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 distance of the corpuscle from the centre of the first sphere, and as the samesphere conjunctly, and is therefore the same as if it all proceeded from asingle corpuscle situate in the centre of the sphere, the entire force withwhich all the corpuscles in the second sphere are attracted, that is, withwhich that whole sphere is attracted, will be the same as if that sphereSEC. Xll.] OP NATURAL PHILOSOPHY. 225were attracted by a force issuing from a single corpuscle in the centre ofthe first sphere ; and is therefore proportional to the distance between thecentres of the spheres. Q,.E.D.CASE 4. Let the spheres attract each other mutually, and the force willbe doubled, but the proportion will remain. Q..E.D.CASE 5. Let the corpuscle p be placed within ^- ^。Ethe sphere AEBF ; and because the force of theplane ef upon the corpuscle is as the solid contained under that plane and the distance jog ; and thecontrary force of the plane EF as the solid contained under that plane and the distance joG ; 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 thedifference of the distances ;that is, as that sum drawn into joS, the distanceof 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, thatis, the attraction of the whole sphere, is conjunctly as the sum of all theplanes, or as the whole sphere, and as joS, the distance of the corpuscle fromthe centre of the sphere. Q.E.D.CASE 6. And if there be composed a new sphere out of innumerable corpuscles such as jo, situate within the first sphere AEBF, it may be proved,as before, that the attraction, whether single of one sphere towards theother, or mutual of both towards each other, will be as the distance joS ofthe centres. Q, E.D.PROPOSITION LXXVIII. THEOREM XXXVIII.If spheres it* the progress from the centre to the circumference be hoivMtvdissimilar a->id unequable, but similar on every side round about af allgiven distances from the centre ; and the attractive force of evsrt/point be as the distance of the attracted body ; I say, that the entireforce with which two spheres of this kind attract each other mutitallijis proportional to the distance between the centres of the spheres.This is demonstrated from the foregoing Proposition, in the same manner as Proposition LXXVI was demonstrated from Proposition LXXY.COR. Those things that were above demonstrated in Prop. X and LXJV,of the motion of bodies round the centres of conic sections, take place whenall the attractions are made by the force of sphaerical bodies of the condition 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 distancesr increase in a simple ratio of the distances, causing the bodies in botli15226 THE MATHEMATICAL PRINCIPLES [BoOK 1cases to revolve in conic sections, and composing sphaerical bodies whosecentripetal forces observe the same law of increase or decrease in the recessfrom the centre as the forces of the particles themselves do ; which is vervremarkable. It would be tedious to run over the other cases, whose conclusions are less elegant and important, so particularly as I have donethese. I choose rather to comprehend and determine them all by one general method as follows.LEMMA XXIX.ff about the centre S there be described any circle as AEB, and about thecentre P there be. also described two circles EF, ef, cutting the Jirst inE and e, and the line PS in F and f; and there be let fall to PS theperpendiculars ED, ed ; I say, that if the distance of the arcs EF; efbe supposed to be infinitely diminished, the last ratio of the evanscentlinr Dd to the evanescent line Ff is the same as that of the line PE tothe live PS.For if the line Pe cut the arc EF in q ; and the right line Ee, whichcoincides with the evanescent arc Ee, be produced, and meet the right linePS in T ; and there be let fall from S to PE the perpendicular SG ; then,because of the like triangles DTE, </!>, DES, it will be as Dd to Ee so))T to TE, or DE to ES : and because the triangles, Ee?, ESG (by Lem.VIII, and Cor. 3, Lem. VII) are similar, it will be as Ee to eq or F/soESto SG ; and, ex ceqno, as Dd to Ff so DE to SG ; that is (because of thesimilar triangles PDE; PGS), so is PE to PS. Q.E.D.PROPOSITION LXXIX. THEOREM XXXIX.Suppose a superficies as EFfe to have its breadth infinitely diminished,and to be just vanishing ; and that the same superficies by its revolutionround the axis PS describes a spherical concavo-convex solid, tothe several equnJ particle* of which there tend equal centripetal forces ;I soy, that the force with which thit solid attracts a corpuscle situatein P is in a ratio compounded of the ratio of the solid DE2 X Ff andthe ratio of the force with which the given particle in the place Ffwould attract the same corpuscle.For if we consider, first, the force of the spherical superficies FE whichSEC. xn.j OF NATURAL PHILOSOPHY. 227is generated by the revolution of the arc FE,and is cut any where, as in r, by the line</6,the annular part of the super J cies generatedby the revolution of the arc rE will be as thelineola Dd, the radius of the sphere PE remainiagthe same; as Archimedes has demonstrated in his Book of the Sphere andCylinder. And the force of this superficies exerted in the direction of the lines PEor Pr situate all round in the conical superficies, will be as this annularsuperficies itself; that is as the lineola DC/, or, which is the same, as therectangle under the given radius PE of the sphere and the lineola DC/ ; butthat force, exerted in the direction of the line PS tending to the centre S,will be less in the ratio PI) to PE, and therefore will be as PD X DC/.Suppose now the line DF to be divided into innumerable little equal particles, each of which call DC/, and then the superficies FE will be dividedinto so many equal annuli, whose forces will be as the sum of all the rectangles PD X DC/, that is, as |PF 2 - |PD 2; and therefore as DE-.Let now the superficies FE be drawn into the altitude F/; and the forceof the solid EF/e exerted upon the corpuscle P will be as DE2 X Ff;that is, if the force be given which any given particle as Ff exerts uponthe corpuscle P at the distance PF. But if that force be not given, theforce of the solid EF/e will be as the solid DE2 X Ff and that force notgiven, conjunctly. Q.E.D.PROPOSITION LXXX. THEOREM XL.If to the several equal parts of a sphere ABE described about the centreS there tend equal centripetal forces ; and from the several points I)in the axis of the sphere AB in which a corpuscle, as F, is placed,there be erected the perpendiculars DE meeting the sphere in E, andif in those perpendiculars the lengths DN be taken as the quantityDE2 X PS-, , and as th*force which a particle of the sphere situate in,the axis exerts at the distance PE upon the corpuscle P conjunctly ; ]say, that the inhole force with which the, corpuscle P is attracted towards the sphere is as the area ANB, comprehended under the axis ofthe sphere AB, and the curve line ANB, the locus of the point N.For supposing the construction in the last Lemma and Theorem tostand, conceive the axis of the sphere AB to be divided into innumerableequal particles DC/, and the whole sphere to be divided into so many spherical concavo-convex laminae EF/e / and erect the perpendicular dn. Bythe last Theorem, the force with which the laminas EF/e attracts the corpuscle P is as DE2 X Ff and the force of one particle exerted at the228 THE MATHEMATICAL PRINCIPLES [BOOK I.distance PE or PF, conjunctly.But (by the last Lemma) Dd is toF/ as PE to PS, and therefore F/.is equal to PEF/ is equal to Dd X; and DE2 XDE2 X PSPET~ ;and therefore the force of the la-DE2 X PSmina EF/e is as Do? X PT?~and the force of a particle exerted at the distance PF conjunctly ; that is,by the supposition, as DN X D(/7 or as the evanescent area DNwrf.Therefore the forces of all the lamina) exerted upon the corpuscle P are asall the areas DN//G?, that is, the whole force of the sphere will be as thewhole area ANB. Q.E.D.COR. 1. Hence if the certripetal force tending to the several particlesp)F 2 vx poremain always the same at all distances, and DN be made as ;Jr Jlithe whole force with which the corpuscle is attracted by the sphere is asthe area ANB.COR. 2. If the centripetal force of the particles be reciprocally as theDE2 X PSdistance of the corpuscle attracted by it, and DN be made as - ^^ ,the force with which the corpuscle P is attracted by the whole sphere wil]be as the area ANB.Cor. 3. Jf the centripetal force of the particles be reciprocally as thecube of the distance of the corpuscle attracted by it, and DN be made asT)F 2 y PS---. the force with which the corpuscle is attracted by the wholesphere will be as the area ANB.COR. 4. And universally if the centripetal force tending to the severalparticles of the sphere be supposed to be reciprocally as the quantity V ;DE2 X PSand D5& be made as ^- ; the force with which a corpuscle is at-Jr Jtj Xtracted by the whole sphere will be as the area ANB.PROPOSITION LXXXI. PROBLEM XLI.T/Le things remaining as above, it is required lo measure the areaANB.From the point P let there be drawn the right line PH touching thesphere in H ; and to the axis PAB, letting fall the perpendicular HI,bisect PI in L; and (by Prop. XII, Book II, Elem.) PE2 is equal tfSEC. XII.] OF NATURAL PHILOSOPHY. 229PS3 + SE2 + 2PSD. But becausethe triangles SPH, SHI are alike,SE2 or SH2 is equal to the rectangle PSI, Therefore PE2 is equalto the rectangle contained under PSand PS -f SI + 2SD ; that is, underPS and 2LS + 2SD ; that is, underPS and 2LD. Moreover DE2 isequal to SE2 SD% or SE2LS 2 + 2SLD LD2, that is, 2SLD LD 2 ALB. For LSSE2or LS a SA a(by Prop. VI, Book II, Elem.) is equal to the rectangle ALB. Therefore if instead of DE2 we write 2SLD LD 2 ALB,the quantity- -^-, which (by Cor. 4 of the foregoing Prop.) is as PE xthe length of the ordinate DN, will2SLD x PS LD 2 X PSnow resolve itself into three partsALB xPS ...-TE3rr~ -pfixT" -pE^-v-; whereifinsteadofVwewntthe inverse ratio of the centripetal force, and instead of PE the mean proportional between PS and 2LD, those three parts will become ordinates toso 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 ofthe sphere be reciprocally as the distance;instead of V write PE the distance, then 2PS X LD for PE 2; and DN will become as SL LDny |y Suppose DN equal to its double 2SL LD - r^ 5 an<* 2SLthe given part of the ordinate drawn into the length AB will describe therectangular area 2SL X AB ; and the indefinite part LD, drawn perpendicularly into the same length with a continued motion, in such sort as inits motion one way or another it may either by increasing or decreasing re-LB 2 -LA 2main always equal to the length LD, will describe the area ^ ,that is, the area SL X AB ; which taken from the former area 2SL XAB, leaves the area SL X AE. But the third part----, drawn after thei lit,same manner with a continued motion perpendicularly into the same length,will describe the area of an hyperbola, which subductedfrom the area SL X AB will leave ANB the area sought.Whence arises this construction of the Problem. Atthe points, L, A, B, erect the perpendiculars L/, Act, B6;making Aa equal to LB, and Bb equal to LA. MakingL/ and LB asymptotes, describe through the points a, 6,230 THE MATHEMATICAL PRINCIPLES [BOOK 1the hyperbolic crrve ab. And the chord ba being drawn, will inclose thearea aba equal to the area sought ANB.EXAMPLE 2. If the centripetal force tending to the several particles ofthe sphere be reciprocally as the cube of the distance, or (which is the samePE3thing; as that cube applied to any given plane ; write2PS X LD for PE2; and DN will become as2AS2SL X AS2for V, andAS 2ALB X AS 22PS X LD2LSIPS X LD 2PSthat is (because PS, AS, SI are continually proportional), asALB X SI2LD:LSIIf we draw then these three parts into thlength AB, the first r-pr will generate the area of an hyperbola ; the sec-L-t 。J, ALB X SI . ALB X SIond iSI the area } AB X SI ;the third2Ll^ area-2LA, that is, !AB X SI. From the first subduct the sum of the2LBsecond and third, and there will remain ANB, the area sought. Whencearises this construction of the problem. At the points L, A, S, B, erectthe perpendiculars L/ Aa Ss, Bb, of which suppose Ssequal to SI ; and through the point s, to the asymptotesL/, LB, describe the hyperbola asb meeting theperpendiculars Aa, Bb, in a and b; and the rectangle2ASI, subducted from the hyberbolic area AasbB, will.. ,, . B leave ANB the area sought.EXAMPLE 3. If the centripetal force tending to the several particles ofthe spheres decrease in a quadruplicate ratio of the distance from the parpT^4 _ tides ; write ~|f- for V, then V 2PS + LD for PE, and DN will become___V2SIXSI 2 X ALB2v2SIXThese three parts drawn into the length AB, produce so many areas, viz.J-L2SI 2 X SL . 1x^ into T r LA~~~5otin* V LB V LA; andBS1 2 X ALB . "1 1"VLA 3 v/LB 3And these after due reduction comeforth __SEC. XII.] OF NATURAL PHILOSOPHY. 23。.2SI 3 4 SI 3~oj-pAnd these by subducting the last from the first, become -oT~rTherefore the entire force with ,7hich the corpuscle P is attracted towardsthe centre of the sphere is as-^, that is, reciprocally as PS 3 X PJQ.E.I.By the same method one may determine the attraction of a corpusclesituate within the sphere, but more expeditiously by the following Theorem.PROPOSITION LXXXIL THEOREM XLI.In a sphere described about the centre S with the interval SA, if there betaken SI, SA, SP continually proportional ; ! sat/, that the attraction,of a corpuscle within the sphere in any place I is to its attraction withoutthe sphere in the place P in a ratio compounded of the subduplicateratio of IS, PS, the distances from the centre, and the subduplicateratio of tJie centripetal forces tending to the centre in those places Pand I.