自然哲学的数学原理-50

mediately drawn back, and converted again towards the earth.BOOK III] OF NATURAL PHILOSOPHY.LEMMA I.If APEp represent the earth uniformly dense, marked with the centre C,the poles P, p, and the equator AE; and if about the centre C, withthe radius CP, we suppose the sphere Pape to be described, and Q li todenote tJie plane on which a right line, drawn from the centre of thesun to the centre of the earth, i/isists at right angles ; and furthersuppose that the several particles of the whole exterior earth PapAP^pE,without the height of the said sphere, endeavour to recede towards t/iisside and that side from the plane Q.R, every particle by a force proportional to its distancefrom that plane ; I say, in the first place, thatthe whole force and efficacy of all the particles that are situate in AE,the circle of the equator, and disposed uniformly without the globe,encompassing the same after the manner of a ring, to iclieel the earthabout its centre, is to the whole force and efficacy of as many particlesin that point A of the equator which is at the greatest distance fromthe plane Q,R, to wheel the earth about its centre with a like circularmotion, as I to 2. And that circular motion will be performed aboutan axis lying in the common section of the equator and the plane Q,R.For let there be described from the centre K, with the diameter IL, thesemi-circle INL. Suppose the semi-circumference INL to be dividedinto innumerable equal parts, and from the several parts N to the diameterQI K ML,IL let fall the sines NM. Then the sums of the squares of all the sineaNM will be equal to the sums of the squares of the sines KM, and bothsums together will be equal to the sums of the squares of as many semidiametersKN ; and therefore the sum of the squares of all the sines NMwill be but half so great as the sum of the squares of as many semi-diameters KN.Suppose now the circumference of the circle AE to be divided into thelike number of little equal parts, and from every such part P a perpendicular FG to be let fall upon the plane QK, as well as the perpendicular AH from the point A. Then the force by which the particle F recede*456 THE MATHEMATICAL PRINCIPLES [BOOK IILfrom the plane QR will (by supposition) be as that perpendicular FG ; andthis force multiplied by the distance CG will represent the power of theparticle F to turn the earth round its centre. And, therefore, the powerof a particle in the place F will be to the power of a particle in the placeA as FG X GO to AH X HC ; that is, as FC 2 to AC2: and thereforethe whole power of all the particles F, in their proper places F, will be tothe power of the like number of particles in the place A as the sum of allthe FC 2 to the sum of all the AC2, that is (by what we have demonstratedbefore), as 1 to 2. Q.E.D.And because the action of those particles is exerted in the direction oflines perpendicularly receding from the plane QR, and that equally fromeach side of this plane, they will wheel about the circumference of the circleof the equator, together with the adherent body of the earth, round an axiswhich lies as well in the plane QR as in that of the equator.LEMMA II.The same things still supposed, I say, in the second place, that the totalforce or poiver of all the particles situated every where about the sphereto turn the earth about the said axis is to the whole force of the likenumber ofparticles, uniformly disposed round the whole circumference,of the equator AE in the fashion of a ring, to turn the whole earthabout with the like circular motion, as 2 to 5.For let IK be any lesser circle parallel tothe equator AE, and let L/ be any two equalparticles in this circle, situated without thesphere Pape ; and if upon the plane QR,which is at right angles with a radius drawnto the sun. we let fall the perpendiculars LM,Im, the total forces by which these particlesrecede from the plane QR will be proportional to the perpendiculars LM, Im. Letthe right line LZ be drawn parallel to theplane Papc, and bisect the same in X ; andthrough the point X draw Nw parallel to the plane QR, and meeting theperpendiculars LM, Im, in N and n and upon the plane QR let fall theperpendicular XY. And the contrary forces of the particles L and I towheel about the earth contrariwise are as LM X MC, and Im X mC ; thatis, as LN X MC + NM X MC, and In X mC nm X mG or LN XMC + NM X MC, and LN x mC NM X mC, and LN X MmNM X MC" 4- raC, the difference of the two, is the force of both takentogether to turn the earth round. The affirmative part of this differenceLN X MA/?,, or 2LN X NX7 is to 2AH X HC, the force of two particlesof the same size situated in A, as LX2 to AC 2; and the negative part NMBOOK 111. OP NATURAL PHILOSOPHY. 45?X MC T wC^or 2XY X CY, is to 2AH X HC, the force of the sametwo particles situated in A, as CX 2 to AC 2. And therefore the differenceof the parts, that is, the force of the two particles L and /, taken together,to wheel the earth about, is to the force of two particles, equal to theformer and situated in the place A, to turn in like manner the earth round,as LX2 CX2 to AC 2. But if the circumference IK of the circle IKis supposed to be divided into an infinite number of little equal parts L,all the LX2 will be to the like number of IX 2 as 1 to 2 (by Lem. 1) ; andto the same number of AC 2 as IX 2 to 2AC2; and the same number olCX2 to as many AC2 as 2CX 2 to 2AC 2. Wherefore the united forcetof all the particles in the circumference of the circle IK are to the jointforces of as many particles in the place A as IX 2 2CX2 to 2AC 2; andtherefore (by Lem. 1) to the united forces of as many particles in the circumference of the circle AE as IX 2 2CX 2 to AC 2.Now if Pp. the diameter of the sphere, is conceived to be divided intoan infinite number of equal parts, upon which a like number of circlesIK are supposed to insist, the matter in the circumference of every circleK will be as IX2; and therefore the force of that matter to turn theearth about will be as IX 2 into IX 2 2CX2: and the force of the samematter, if it was situated in the circumference of the circle AE, would be asIX 2 into AC 2. And therefore the force of all the particles of the wholematter situated without the sphere in the circumferences of all the circle?is to the force of the like number of particles situated in the circumference of the greatest circle AE as all the IX 2 into IX2 2CX 2 to asmany IX 2 into AC2;that is, as all the AC 2 CX2 into AC2 3CX 2to as many AC2 CX2 into AC2: that is, as all the AC 4 4AC 2 xCX2 + 3CX4 to as many AC 4 AC 2 X CX2;that is, as the wholefluent quantity, whose fluxion is AC 4 4AC 2 X CX 3 + 3CX 4, to thewhole fluent quantity, whose fluxion is AC 4 AC 2 X CX 2; and, therefore, by the method of fluxions, as AC 4 X CX fAC 2 X CX 3 +|CX. 5 to AC 4 X CX i-AC2 X CX3;that is, if for CX we write thewhole Cp, or AC, as T4jAC 5 to fAC 5;that is, as 2 to 5. Q.E.D.LEMMA III.The same things still supposed, I say, in the third place, that the motion of the i^hole earth about the axis above-named arising from themotions of all the particles, will be to the motion of the aforesaid ringabout the same axis in a, proportion compounded of the proportion ofthe matter in the earth to the matter in the ring ; and the proportionof three squares of the quadrantal arc of any circle to two squaresof its diameter, that is, in the proportion of the matter to the matter,and of ttie number 925275 to the number 1000000.the motion of a cylinder revolved about its quiescent axis is to the*68 THE MATHEMATICAL PRINCIPLES [BOOK III.motion of the inscribed sphere revolved together with it as any four equalsquares to three circles inscribed in three of those squares ; and the motion of this cylinder is to the motion of an exceedingly thin ring surrounding both sphere and cylinder in their common contact as double thematter in the cylinder to triple the matter in the rir^j ; and this motionof the ring, uniformly continued about the axis of the cylinder, is to theuniform motion of the same about its own diameter performed in thesame periodic time as the circumference of a circle to double its diameter.HYPOTHESIS II.If the other parts of the earth were taken away, and the remaining ringwas carried alone about the sun in, the orbit of the earth by the annualmotion, while by the diurnal motion it ivas in the mean time revolvedabout its own axis inclined to the plane of t/te ecliptic by an angleof 23i decrees, the motion of the equinoctial points would be thesame, whether the ring were fluid, or whether it consisted of a hardand rigid matter.PROPOSITION XXXIX. PROBLEM XX.ToJind the precession of the equinoxes.The middle horary motion of the moon s nodes in a circular orbit, whenthe nodes are in the quadratures, was 16" 35 " 16iv. 36V.; the half ofwhich, 8" 17 " 38 v. 18V. (for the reasons above explained) is the mean horary motion of the nodes in such an orbit, which motion in a whole sidereal year becomes 20 11 46". Because, therefore, the nodes of the moonin such an orbit would be yearly transferred 20 11 46" in antecederttia ;and, if there were more moons, the motion of the nodes of every one (byCor. 16, Pro]). LXVI. Book 1) would be as its periodic time; if upon thesurface of the earth a moon was revolved in the time of a sidereal day,the annual motion of the nodes of this moon would be to 20 31 46" as23h. 56 , the sidereal day, to 27 !. 7h. 43 , the periodic time of our moon,that is, as 1436 to 39343. And the same thing would happen to thenodes of a ring of moons encompassing the earth, whether these moonsdid not mutually touch each the other, or whether they were molten, andformed into a continued ring, or whether that ring should become rigidand inflexible.Let us, then, suppose that this ring is in quantity of matter equal tothe whole exterior earth PctpAPepR, which lies without the sphere Pape(see fig. Lem. II) ; and because this sphere is to that exterior earth as CtoAC 2 aC2, that is (seeing PC or C the lea^t semi-diameter of theearth is to AC the greatest semi-diameter of the same as 229 to 230), as52441 to 459 : if this ring encompassed the earth round the equator, andboth together were revolved about the diameter of the ring, the motion ofHOOK III.] OF NATURAL PHILOSOPHY. 459the ring (by Lcm. Ill) would be to the motion of the inner sphere as 459to 52441 and 1000000 to 925275 conjunct!}, that is, as 4590 to 485223;and therefore the motion of the ring would be to the sum of the motionsof both ring and sphere as 4590 to 489813. Wherefore if the ring adheres to the sphere, and communicates its motion to the sphere, by whichits nodes or equinoctial points recede, the motion remaining in the ring willbe to its former motion as 4590 to 489813; upon which account themotion of the equinoctial points will be diminished in the same proportion. Wherefore the annual motion of the equinoctial points of the body,composed of both ring and sphere, will be to the motion 20 11 46" as1436 to 39343 and 4590 to 489813 conjunctly, that is, as 100 to 292369.But the forces by which the nodes of a number of moons (as we explainedabove), and therefore by which the equinoctial points of the ring recede(that is, the forces SIT, in fig. Prop. XXX), are in the several particlesas the distances of those particles from the plane Q,R ; and by these forcesthe particles recede from that plane : and therefore (by Lem. II) if thematter of the ring was spread all over the surface of the sphere, after thefashion of the figure PupAPepl^, in order to make up that exterior partof the earth, the total force or power of all the particles to wheel aboutthe earth round any diameter of the equator, and therefore to move theequinoctial points, would become less than before in the proportion of 2 to5. Wherefore the annual regress of the equinoxes now would be to 2011 46" as 10 to 73092 ; that is. would be 9" 56 " 50iv.But because the plane of the equator is inclined to that of the ecliptic,this motion is to be diminished in the proportion of the sine 91706(which is the co-sine of 23 1 deg.) to the radius 100000 ; and the remaining motion will now be 9" 7 " 20iv. which is the annual precession of theequinoxes arising from the force of the sun.But the force of the moon to move the sea was to the force of the sunnearly as 4,4815 to 1; and the force of the moon to move the equinoxesis to that of the sun in the same proportion. Whenoe the annual precessionof the equinoxes proceeding from the force of the moon comes out 40"52" 521V. and the total annual precession arising from the united forcesof both will be 50" 00" 12iv. the quantity of which motion agrees withthe phaenomena ; for the precession of the equinoxes, by astronomical observations, is about 50" yearly.If the height of the earth at the equator exceeds its height at thepoles by more than 17| miles, the matter thereof will be more rare nearthe surface than at the centre; and the precession of the equinoxes willbe augmented by the excess of height, and diminished by the greater rarity,And now we have described the system of the sun, the earth, moon,and planets, it remains that we add something about the comets.460 THE MATHEMATICAL PRINCIPLES [BOOK IILLEMMA IVThat the comets are higher tliau tJie moon, and in the regions of theplanets.As the comets were placed by astronomers above the moon, because theywere found to have no diurnal parallax, so their annual parallax is a convincing proof of their descending into the regions of the planets ; for allthe comets which move in a direct course according to the order of thesigns, about the end of their appearance become more than ordinarily slowor retrograde, if the earth is between them and the sun ; and more thanordinarily swift, if the earth is approaching to a heliocentric oppositionwith them ; whereas, on the other hand, those which move against the order of the signs, towards the end of their appearance appear swifter thanthey ought to be, if the earth is between them and the sun ; and slower,and perhaps retrograde, if the earth is in the other side of its orbit. Andthese appearances proceed chiefly from the diverse situations which theearth acquires in the course of its motion, after the same manner as it happens to the planets, which appear sometimes retrograde, sometimes moreslowly, and sometimes more swiftly, progressive, according as the motion ofthe earth falls in with that of the planet, or is directed the contrary wav.If the earth move the same way with the comet, but, by an angular motionabout the sun, so much swifter that right lines drawn from the earth tothe comet converge towards the parts beyond the comet, the comet seenfrom the earth, because of its slower motion, will appear retrograde ; andeven if the earth is slower than the comet, the motion of the earth beingsubducted, the motion of the comet will at least appear retarded; but if theearth tends the contrary way to that of the cornet, the motion of the cometwill from thence appear accelerated; and from this apparent acceleration,or retardation, or regressive motion, the distance of the comet may be in-F c B A ferred in this manner. Let TQA,TQ,B, TQ,C, be three observed longitudes of the comet about the timeof its first appearing, and TQ,F itslast observed longitude before itsdisappearing. Draw the right lineABC, whose parts AB, BC, interceptedbetween the right lines QAand Q.B, QB and Q.C, may be one to the other as the two times betweenthe three first observations. Produce AC to G, so as AG may be to ABas the time between the first and last observation to the time between thefirst and second ; and join Q.G. Now if the comet did move uniformly ina right line, and the earth either stood still, or was likewise carried foruards in a right line by an uniform motion, the angle TQG would be thtBOOK 111.] OF NATURAL PHILOSOPHY. 401longitude of the comet at the time of the last observation. The angle,therefore, FQG, which is the difference of the longitude, proceeds from theinequality of the motions of the comet and the earth ; and this angle, ifthe earth and cornet move contrary ways, is added to the angle TQ,G, andaccelerates the apparent motion of the comet ; but if the comet move thesame way with the earth, it is subtracted, and either retards the motion olthe comet, or perhaps renders it retrograde, as we have but now explained.This angle, therefore, proceeding chiefly from the motion of the earth, isjustly to be esteemed the parallax of the comet; neglecting, to wit, somelittle increment or decrement that may arise from the unequal motion ofthe comet in its orbit : and from this parallax we thus deduce the distanceof the comet. Let S represent the sun, acT vthe orbis tnagnus, a the earth s place in thefirst observatiun, c the place of the earth inthe third observation, T the place of theearth in the last observation, and TT a rightline drawn to the beginning of Aries. Setoff the angle TTV equal to the angle TQF,that is, equal to the longitude of the cometat the time when the earth is in T ; join ac,and produce it to g1, so as ag may be to acas AG to AC ; and g will be the place atwhich the earth would have arrived in thetime of the last observation, if it had continued to move uniformly in the right lineac. Wherefore, if we draw g T parallel to TT, and make the angle T^Vequal to the angle TQ,G, this angle Tg。 will be equal to the longitude ofthe comet seen from the place g, and the angle TVg- will be the parallaxwhich arises from the earth s being transferred from the place g into theplace T ; and therefore V will be the place of the comet in the plane of theecliptic. And this place V is commonly lower than the orb of Jupiter.The same thing may be deduced from the incurvation of the way of thecomets ; for these bodies move almost in great circles, while their velocityis great ; but about the end of their course, when that part of their apparent motion which arises from the parallax bears a greater proportion totheir whole apparent motion, they commonly deviate from those circles, andwhen the earth goes to one side, they deviate to the other : and this deflexion, because of its corresponding with the motion of the earth, must arisechiefly from the parallax ; and the quantity thereof is so considerable, as,by my computation, to place the disappearing comets a good deal lowerthan Jupiter. Whence it follows that when they approach nearer to us intheir perigees and perihelions they often descend below the orbs of Mareand the inferior planets.462 THE MATHEMATICAL PRINCIPLES [BOOK JJI,The near approach of the comets is farther confirmed from the light ofr heads; for the light of a celestial body, illuminated by the sun, andreceding to remote parts, is diminished in the quadruplicate proportion ofthe distance; to wit, in one duplicate proportion, on account of the increaseof the distance from the sun, and in another duplicate proportion, on account of the decrease of the apparent diameter. Wherefore if both thequantity of light and the apparent diameter of a comet are given, its distance will be also given, by taking the distance of the comet to the distanceof ;i planet in the direct proportion of their diameters and the reciprocalsubduplicate proportion of their lights. Thus, in the comet of the year1682, Mr. Flamsted observed with a telescope of 16 feet, and measuredwith a micrometer, the least diameter of its head; 2 00; but the nucleusor star in the middle of the head scarcely amounted to the tenth part ofthis measure; and therefore its diameter was only 11" or 12" but in thelight and splendor of its head it surpassed that of the comet in the year1680; and might be compared with the stars of the lirst or second magnitude. Let us suppose that Saturn with its ring was about four times morelucid; and because the light of the ring was almost equal to the light ofthe globe within, and the apparent diameter of the globe is about 21", andtherefore the united light of both globe and ring would be equal to thelight of a globe whose diameter is 30", it follows that the distance of thcomet was to the distance of Saturn as 1 to v/4 inversely, and 12" to 30directly ; that is, as 24 to 30, or 4 to 5. Again ; the comet in the monthof April 1665, as Hevelius informs us, excelled almost all the fixed starsin splendor, and even Saturn itself, as being of a much more vivid colour ;for this comet was more lucid than that other which had appeared aboutthe end of the preceding year, and had been compared to the stars of thehrst magnitude. The diameter of its head was about 6; but the nucleus,compared with the planets by means of a telescope, was plainly less thanJupiter ; and sometimes judged less, sometimes judged equal, to the globeof Saturn within the ring. Since, then, the diameters of the heads of thecomets seldom exceed 8 or 12;, and the diameter of the nucleus or centralstar is but about a tenth or perhaps fifteenth part of the diameter of thehead, it appears that these stars are generally of about the same apparentmagnitude with the planets. But in regard that their light may be oftencompared with the light of Saturn, yea, and sometimes exceeds it, it is evident that all comets in their perihelions must either be placed below or notfar above Saturn ; and they are much mistaken who remove them almostas far as the fixed stars ;for if it was so, the comets could receive no morelight from our sun than our planets do from the fixed stars.So far we have gone, without considering the obscuration which cometssuffer from that plenty of thick smoke which encompasseth their heads,and through which the heads always shew dull, as through i cloud; for byBoOK Hl.J Or NATURAL PHILOSOPHY. 463how much the more a body is obscured by this smoke, by so much the morenear it must be allowed to come to the sun, that it may vie with the planetain the quantity of light which it reflects. Whence it is probable thatthe comets descend far below the orb of Saturn, as we proved before froutheir parallax. But, above all, the thing is evinced from their tails, whichmust be owing either to the sun s light reflected by a smoke arising fromthem, and dispersing itself through the aether, or to the light of their own