自然哲学的数学原理-15

diameter. Let AGOF be an attracting spheroid, S its centre, and P the body attracted.Through the body P let there be drawn thesemi-diameter SPA, and two right lines DE,FC meeting the spheroid in 1) and E, F andG ; and let PCM, HLN be the superficies of240 THE MATHEMATICAL PRINCIPLE* ffioOK 1.two interior spheroids similar and concentrical to the exterior, the first ofwhich passes through the body P. and cuts the right lines DE, FG in Band C ; arid the latter cuts the same right lines in H and I, K and L.I ,et the spheroids have all one common axis, and the parts of the rightlines intercepted on both sides DP and BE, FP and CG, DH and IE, FKand LG, will be mutually equal; because the right lines DE. PB, and HI.are bisected in the same point, as are also the right lines FG, PC, and KL.Conceive now DPF. EPG to represent opposite cones described with theinfmitely small vertical angles DPF, EPG, and the lines DH, El to beinfinitely small also. Then the particles of the cones DHKF, GLIE, cutoff by the spheroidical superficies, by reason of the equality of the lines DHand ET; will be to one another as the squares of the distances from the bodyP, and will therefore attract that corpuscle equally. And by a like reasoning if the spaces DPF, EGCB be divided into particles by the superficies of innumerable similar spheroids concentric to the former and having J. Oone common axis, all these particles will equally attract on both sides thebody P towards contrary parts. Therefore the forces of the cone DPF.and of the conic segment EGCB, are equal, and by their contrariety destroy each other. And the case is the same of the forces of all the matterthat lies without the interior spheroid PCBM. Therefore the body P isattracted by the interior spheroid PCBM alone, and therefore (by Cor. 3,Prop. 1 ,XXII) its attraction is to the force with which the body A is attracted by the whole spheroid AGOD as the distance PS to the distanceAS. Q.E.D.PROPOSITION XCII. PROBLEM XLVI.An attracting body being given, it is required to find the ratio of the decrease of the centripetalforces tending to its several points.The body given must be formed into a sphere, a cylinder, or some regular figure, whose law of attraction answering to any ratio of decrease maybe found by Prop. LXXX, LXXXI, and XCI. Then, by experiments,the force of the attractions must be found at several distances, and the lawof attraction towards the whole, made known by that means, will givethe ratio of the decrease of the forces of the several parts ; which was tobe found.PROPOSITION XCIII. THEOREM XLVII.If a solid be plane on one side, and infinitely extended on all otljer sides,and consist of equal particles equally attractive, ivhose forces decrease,in the recessfrom the solid, in the ratio of any power greater than thesquare of the distances ; and a corpuscle placed towards eitfar part ofthe plane is attracted by the force of the whole solid ; I say that tfieattractive force of the whole solid, in the recessfrom its platw superfiXIILj OF NATURAL PHILOSOPHY". 241nHmGties, will decrease in the ratio of a power whose side is the distance ojthe corpuscle from the plane, and its index less by 3 than the index ojthe power of the distances.CASE 1. Let LG/be the plane by whichthe solid is terminated. Let the solidlie on that hand of the plane that is towards I, and let it be resolved into in- _.numerable planes mHM, //IN, oKO,(fee., parallel to GL. And first let theattracted body C be placed without thesolid. Let there be drawn CGHI perpendicular to those innumerable planes,and let the attractive forces of the points of the solid decrease in the ratioof a power of the distances whose index is the number n not less than 3.Therefore (by Cor. 3, Prop. XC) the force with which any plane mHMattracts the point C is reciprocally as CHn 2. In the plane mHM take thelength HM reciprocally proportional to CH 1 2, and that force will be asHM. In like manner in the several planes /GL, //,TN, oKO, (fee., take thelengths GL, IN, KO, (fee., reciprocally proportional to CGn 2, CI 1 2,CKn 2, (fee., and the forces of those planes will bs as the lengths so taken,and therefore the sum of the forces as the sum of the lengths, that is, theforce of the Avhole solid as the area GLOK produced infinitely towardsOK. But that area (by the known methods of quadratures) is reciprocallyas CGn 3, and therefore the force of the whole solid is reciprocally asCG"-3. Q.E.D.CASE 2. Let ttecorpuscleC be now placed on thathand of the plane /GL that is within the solid,and take the distance CK equal to the distanceCG. And the part of the solid LG/oKO terminated by the parallel planes /GL, oKO, will attract the corpuscle C, situate in the middle, neitherone way nor another, the contrary actions of theopposite points destroying one another by reason oftheir equality. Therefore the corpuscle C is attracted by the force onlyof the solid situate beyond the plane OK. But this force (by Case 1) isreciprocally as CKn 3, that is, (because CG, CK are equal) reciprocally asCG"3. Q,.E.D.COR. 1. Hence if the solid LGIN be terminated on each sitfe by two infinite parallel places LG, IN, its attractive force is known, subductingfrom the attractive force of the whole infinite solid LGKO the attractiveforce of the more distant part NIKO infinitely produced towards KO.COR. 2. If the more distant part of this solid be rejected, because its attraction compared with the attraction of the nearer part is inconsiderable,16242 THE MATHEMATICAL PRINCIPLES [BOOK 1the attraction of that nearer part will, as the distance increases, decreasenearly in the ratio of the power CGn 3.Con. 3. And hence if any finite body, plane on one side, attract a corpuscle situate over against the middle of that plane, and the distance betweenthe corpuscle and the plane compared with the dimensions of the attractingbody be extremely small;and the attracting body consist of homogeneousparticles, whose attractive forces decrease in the ratio of any power of thedistances greater than the quadruplicate ; the attractive force of the wholebody will decrease very nearly in the ratio of a power whose side is thatvery small distance, and the index less by 3 than the index of the formerpower. This assertion does not hold good, however, of a body consistingof particles whose attractive forces decrease in the ratio of the triplicatepower of the distances ; because, in that case, the attraction of the remoterpart of the infinite body in the second Corollary is always infinitely greaterthan the attraction of the nearer part.SCHOLIUM.If a body is attracted perpendicularly towards a given plane, and fromthe law of attraction given, the motion of the body be required ; the Problem will be solved by seeking (by Prop. XXXIX) the motion of the bodydescending in a right line towards that plane, and (by Cor. 2, of the Laws)compounding that motion with an uniform motion performed in the direction of lines parallel to that plane. And, on the contrary, if there be required the law of the attraction tending towards the plane in perpendicular directions, by which the body may be caused to move in any givencurve line, the Problem will be solved by working after the manner of thethird Problem.But the operations may be contracted by resolving the ordinates intoconverging series. As if to a base A the length B be ordinately applied in any given angle, and that length be as any power of the baseA^ ; and there be sought the force with which a body, either attracted towards the base or driven from it in the direction of that ordinate, may becaused to move in the curre line which that ordinate always describes withits superior extremity ;I suppose the base to be increased by a very small,m mpart O, and I resolve the ordinate A -f Ol^ into an infinite series A- -f!!L OA^+ ^-^--- OOA ;- &c., and I suppose the force proper-11 1111tional to the term of this series in which O is of two dimensions, that is,to the term - - OOA ^YT, Therefore the force sought is aaSEC. XIV.J OF NATURAL PHILOSOPHY. 2Mmm mn m 2n .... mm mn m 2n A 7, , or, which is the same thinor, as L> m .nn nnAs if the ordinate describe a parabola, m being 2, and n = 1, the forcewill be as the given quantity 2B, and therefore is given. Therefore witha given force the body will move in a parabola, as Galileo has demonstrated. If the ordinate describe an hyperbola, m being = 1, and n1, the force will be as 2A 3 or 2B 3; and therefore a force which is as thecube of the ordinate will cause the body to move in an hyperbola. Butleaving this kind of propositions, I shall go on to some others relating tomotion which I have fiot yet touched upon.SECTION XIV.Of the motion of very small bodies when agitated by centripetal forcestending to the several parts of any very great body.PROPOSITION XCIV. THEOREM XLVIII.If two similar mediums be separatedfrom each other by a space terminated on both sides by parallel planes, and a body in its passagethrough that space be attracted or impelled perpendicularly towardseither of those mediums, and not agitated or hindered by any otherforce ; and the attraction be every where the same at equal distancesfrom either plane, taken towards the same hand of the plane ; I say,that the sine of incidence upon either plane will be to the sine of emcrgencefrom the other plane in a given ratio.CASE 1. Let Aa and B6 be two parallel planes,and let the body light upon the first plane Aa inthe direction of the line GH, and in its wholepassage through the intermediate space let it beattracted or impelled towards the medium of incidence, and by that action let it be made to describe a curve line HI, and let it emerge in the direction of the line IK. Let there be erected IMperpendicular to Eb the plane of emergence, andmeeting the line of incidence GH prolonged in M, and the plane of incidence Aa in R ; and let the line of emergence KI be produced and meetHM in L. About the centre L, with the interval LI, let a circle be described cutting both HM in P and Q, and MI produced in N ; and, first,if the attraction or impulse be supposed uniform, the curve HI (by whatGalileo has demonstrated) be a parabola, whose property is that of a rocTHE MATHEMATICAL PRINCIPLES [BoOK 1tangle under its given latiis rectum and the line IM is equal to the squarrfcf HM ; and moreover the line HM will be bisected in L. Whence if toMI there be let fall the perpendicular LO, MO, OR will be equal; andadding the equal lines ON, OI, the wholes MN, IR will be equal also.Therefore since IR is given, MN is also given, and the rectangle NMI isto the rectangle under the latus rectum and IM, that is, to HMa in a givenratio. But the rectangle NMI is equal to the rectangle PMQ,, that is, tothe difference of the squares ML2, and PL2 or LI2; and HM2 hath a givenratio to its fourth part ML2; therefore the ratio of ML2 LI2 to ML2is given,and by conversion the ratio of LI2 to ML , and its subduplicate, theratrioof LI to ML. But in every triangle, as LMI, the sines jf the angles areproportional to the opposite sides. Therefore the ratio of the sine of theangle of incidence LMR to the sine of the angle of emergence LIR isgiven. QJE.lr).CASE 2. Let now the body pass successively through several spaces terminated with parallel planes Aa/>B, B6cC, &c., and let it be acted on by a。 . force which is uniform in each of them separ-。 a ately, but different in the different spaces ; andB 。 fr by what was just demonstrated, the sine of thec ^^ c angle of incidence on the first plane Aa is tothe sine of emergence from the second plane Bbin a given ratio; and this sine of incidence upon the second plane Bb willbe to the sine of emergence from the third plane Cc in a given ratio; andthis sine to the sine of emergence from the fourth plane Dd in a given ratio; and so on in infinitum ; and, by equality, the sine of incidence onthe first plane to the sine of emergence from the last plane in a given ratio.I ,et now the intervals of the planes be diminished, and their number be infinitely increased, so that the action of attraction or impulse, exerted according to any assigned law, may become continual, and the ratio of the sine ofincidence on the first plane to the sine of emergence from the last planebeing all along given, will be given then also. QJE.D.PROPOSITION XCV. THEOREM XLIX.The same things being supposed, I say, that the velocity of the body before its incidence is to its velocity after emergence as the sine of emergence to the sine of incid nee.Make AH and Id equal, and erect the perpendicularsAG, dK meeting the lines of incidenceand emergence GH, IK, in G and K. In GH-- take TH equal to IK, and to the plane Aa let^ fall a perpendicular TV. And (by Cor. 2 of the|x^ ILaws of Motion) let the motion of the body bej v- resolved into two, one perpendicular to the planesSEC. X1V.J OF NATURAL PHILOSOPHY. 245Aa, Bb, Cc, &c, and another parallel to them. The force of attraction orimpulse, acting in directions perpendicular to those planes, does not at allalter the motion in parallel directions ; and therefore the body proceedingwith this motion will in equal times go through those equal parallel intervals that lie between the line AG and the point H, and between the pointI and the line dK ;that is, they will describe the lines GH, IK in equaltimes. Therefore the velocity before incidence is to the velocity afteremergence as GH to IK or TH, that is, as AH or Id to vH, that is (supposing TH or IK radius), as the sine of emergence to the sine of incidence. Q.E.D.PROPOSITION XCVL THEOREM L.The same things being supposed, and that the motion before incidence isswifter than afterwards ; 1 sat/, lhat if the line of incidence be inclined continually, the body will be at last reflected, and the angle ofreflexion will be equal to the angle of incidence.For conceive the body passing between the parallel planes Aa, Bb, Cc,&c., to describe parabolic arcs as above;and let those arcs be HP, PQ, QR, &c.And let the obliquity of the line of inci- gdence GH to the first plane Aa be such rc~that the sine of incidence may be to the radius of the circle whose sine it is,in the same ratio which the same sine of incidence hath to the sine of emergence from the plane Dd into the space DefeE ; and because the sine ofemergence is now become equal to radius, the angle of emergence will be aright one, and therefore the line of emergence will coincide with the planeDd. Let the body come to this plane in the point R ; and because theline of emergence coincides with that plane, it is manifest that the body canproceed no farther towards the plane Ee. But neither can it proceed in theline of emergence Rd; because it is perpetually attracted or impelled towardsthe medium of incidence. It will return, therefore, between the planes Cc,Dd, describing an arc of a parabola Q,R</, whose principal vertex (by whatGalileo has demonstrated) is in R, cutting the plane Or in the same angleat q, that it did before at Q, ; then going on in the parabolic arcs qp, ph,&c., similar and equal to the former arcs QP, PH, &c., it will cut the restof the planes in the same angles at p, h, (fee., as it did before in P, H, (fee.,and will emerge at last with the same obliquity at h with which it firstimpinged on that plane at H. Conceive now the intervals of the planesAa, Bb, Cc, Dd, Ee, (fee., to be infinitely diminished, and the number infinitely increased, so that the action of attraction or impulse, exerted according to any assigned law, may become continual; and, the angle ofemergence remaining all alor g equal to the angle of incidence, will beequal to the same also at last. Q.E.D.246 THE MATHEMATICAL PRINCIPLES IBoOK 1SCHOLIUM.These attractions bear a great resemblance to the reflexions and refractions of light made in a given ratio of the secants, as was discovered h}Siiellius ; and consequently in a given ratio of the sines, as was exhibitedby Hes Cortes. For it is now certain from the phenomena of Jupiter s^satellites, confirmed by the observations of different astronomers, that lightis propagated in succession, and requires about seven or eight minutes totravel from the sun to the earth. Moreover, the rays of light that are inour air (as lately was discovered by Grimaldus, by the admission of lightinto a dark room through a small hole, which 1 have also tried) in theirpassage near the angles of bodies, whether transparent or opaque (such aathe circular and rectangular edges of gold, silver and brass coins, or ofknives, or broken pieces of stone or glass), are bent or inflected round thosebodies as if they were attracted to them ; and those rays which in theirpassage come nearest to the bodies are the most inflected, as if they weremost attracted : which tiling I myself have also carefully observed. Andthose which pass at greater distances are less inflected; and those at stillgreater distances are a little inflected the contrary way, and form threefringes of colours. In the figure 5 represents the edge of a knife, or any-f:::r; N ^c :.-/ >V V U JW~"~a~" "~a C: O lakind of wedge AsB : and gowog,fmnif,emtme, dlsld, are rays inflected towards the knife in the arcs owo, nvn, mtm, Isl ; which inflection is greateror less according; to their distance from the knife. Now since this inflection of the rays is performed in the air without the knife, it follows that therays which fall upon the knife are first inflected in the air before they touchthe knife. And the case is the same of the rays falling upon glass. Therefraction, therefore, is made not in the point of incidence, but gradually, bya continual inflection of the rays ; which is done partly in the air before theytouch the glass, partly (if [ mistake not) within the glass, after they haveentered it;as is represented in the rays ckzc, bujb^ ahxa, falling upon r,q, p, and inflected between k and z, i and y, h and x. Therefore becauseof the analogy there is between the propagation of the rays f light and themotion of bodies, I thought it not amiss to add the followi g Propositionsfar optical uses ; not at all. considering the nature of the rays of .light, orinquiring whether they are bodies or not ; but only determining the trajectories of bodies which are extremely like the trajectories of the rays.SEC. XIV.] OF NATURAL PHILOSOPHY. 247PROPOSITION XCVII. PROBT.-EM XLVII.Supposing t/w sine of incidence upon any superficies to be in a given ratio to the sine of emergence ; and that tha inflection of t/ts paths ofthose bodies near that superficies is performed in a very short space,which may be considered as a point ; it is required to determine sucka superficies as may cause all the corpuscles issuing from any onegiven place to converge to another given place.Let A be the place from whence the cor- Epuscles diverge ; B the place to which theyshould converge ; CDE the curve line whichby its revolution round the axis AB describes . /Cthe superficies sought ; D, E, any two points of that curve ; and EF, EG,perpendiculars let fall on the paths of the bodies AD, DB. Let the pointD approach to and coalesce with the point E ; and the ultimate ratio ofthe line DF by which AD is increased, to the line DG by which DB isdiminished, will be the same as that of the sine of incidence to the sine ofemergence Therefore the ratio of the increment of the line AD to thedecrement of the line DB is given: and therefore if in the axis AB therebe taken any where the point C through which the curve CDE mustpass, and CM the increment of AC be taken in that given ratio to CNthe decrement of BC, and from the centres A, B, with the intervals AM,BN, there be described two circles cutting each other in D ; that point Dwill touch the curve sought CDE, and, by touching it any where at pleasure,will determine that curve. Q.E.I.COR. 1. By causing the point A or B to go off sometimes in infinitum,and sometimes to move towards other parts of the point C, will bo obtained all those figures which Cartesins has exhibited in his Optics and Geometry relating to refractions. The invention of which Cartcsius havingthought fit to conceal, is here laid open in this Proposition.COR. 2. If a body lighting on any superficies CD in the direction of a riO^ht line AD, Qj- 。drawn according to any law, should emergein the direction of another right line DK ;and from the point C there be drawn curvelines CP, CQ, always perpendicular to AD, DK ; the increments of thelines PD, QD, and therefore the lines themselves PD, Q.D, generated bythose increments, will be as the sines of incidence and emergence to eachother, and e contra.PROPOSITION XCVIII. PROBLEM XLVIII.The same things supposed ; if round the axis AB any attractive superficies be described as CD, regular or irregular, through which the bodies issuing from the given place A must pass ; it is required to findTHE MATHEMATICAL PRINCIPLES. [BOOK Ja second attractive superficies EF, which may make those bodies converge to a given place B.Let a line joining AB cutthe first superficies in C andthe second in E, the point Dbeing taken any how at pleasure. And supposing thef sine of incidence on the firstsuperficies to the sine ofemergence from the same, and the sine of emergence from the second superficies to the sine of incidence on the same, to be as any given quantity Mto another given quantity N; then produce AB to G, so that BG may beto CE as M N to N ; and AD to H, so that AH may be equal to AG ;arid DF to K, so that DK may be to DH as N to M. Join KB, and aboutthe centre D with the interval DH describe a circle meeting KB producedin L, and draw BF parallel to DL; and the point F will touch the lineEF, which, being turned round the axis AB, will describe the superficiessought. Q.E.F.For conceive the lines CP, CQ to be every where perpendicular to AD,