自然哲学的数学原理-58

orbits (p 403), and, by radii drawn to the sun, describe areas nearly proportional to the times, as is explained in Prop. LXV. If the sun was quiescent, and the other planets did not act mutually one upon another, theirorbits would be elliptic, and the areas exactly proportional to the times (byProp. XI, and Cor. 1, Prop. XIII). But the actions of the planets amonirthemselves, compared with the actions of the sun on the planets, are of nomoment, and produce no sensible errors. And those errors are less in revolutions about the sun agitated in the manner but now described than ifthose revolutions were made about the sun quiescent (by Prop. LXV1, andCor. Prop. LXVIll), especially if the focus of every orbit is placed in thecommon centre of gravity of all the lower included planets; viz., the focusof the orbit of Mercury in the centre of the sun : the focus of the orbit ofVenus in the common centre of gravity of Mercury and the sun ; the focusof the orbit of thp earth in the common centre of gravity of Venus, Mercury, and the sun ; and so of the rest. And by this means the foci of thecrbits of all the planets, except Saturn, will not be sensibly removed fromthe centre of the sun, nor will the focus of the orbit of Saturn recede sensibly from the common centre of gravity of Jupiter and the sun. Andtherefore astronomers are not far from the truth, when they reckon thesun s centre the common focus of all the planetary orbits. In Saturn itselfTHE SYSTEM CF THE W )RLD.the error thence arising docs not exceed 1 45 . And if its orbit, by placingthe focus thereof in the common centre of gravity of Jupiter and the sun,ghall happen to agree better with the phenomena, from thence all that wehave said will be farther confirmed.If the sun was quiescent, and the planets did not act one on another, theaphelions and nodes of their orbits would likewise (by Prop. 1, XI, and Cor.Prop. XIU) be quiescent. And the longer axes of their elliptic orbitswould (by Prop. XV) be as the cubic roots of the squares of their periodictimes : and therefore from the given periodic times would be also given.But those times are to be measured not from the equinoctial points, whichare rnoveable, but from the first star of Aries. Put the semi-axis of theearth s orbit 100000, and the semi-axes of the orbits of Saturn, Jupiter,Mars, Venus, and Mercury, from their periodic times, will come out953806, 520116, 152399, 72333, 38710 respectively. But from the sun smotion every semi-axis is increased (bv Prop. LX) by about one third ofthe distance of the sun s centre from the common centre of gravity ofthe sun and planet (p. 405, 406.) And from the actions of the exteriorplanets on the interior, the periodic times of the interior are somethingprotracted, though scarcely by any sensible quantity ; and their aphelionsare transferred (by Cor. VI and VII, Prop. LXVI)by very slow motionsin conset/ue/ttia. And on the like account the periodic times of all, especially of the exterior planets, will be prolonged by the actions of thesomets, if any such there are, without the orb of Saturn, and the aphelions of all will be thereby carried forwards in consequent-la. But fromthe progress of the aphelions the regress of the nodes follows (by Cor.XI, XIII, Prop. 1 jXVI). And if the plane of the ecliptic is quiescent, theregress of the nodes (by Cor. XVI, Prop. LX.VI) will be to the progress of*he aphelion in every orbit as the regress of the nodes of the moon s orbitto the progress of its apogeon nearly, that is, as about 10 to 21. But astronomical observations seem to confirm a very slow progress of the aphelions, and a regress of the nodes in respect of the fixed stars. And henceit is probable that there are comets in the regions beyond the planets, which,revolving in very eccentric orbs, quickly fly through their perihelion parts,and, by an exceedingly slow motion in their aphelions, spend almost theirwhole time in the regions beyond the planets ;as we shall afterwards explain more at large.The planets thus revolved about the sun (p. 413, 41.4, 415) may at thesame time carry others revolving about themselves as satellites or moons,as appears by Prop. LXVI. But from the action of the sun our moonmust move with greater velocity, and, by a radius drawn to the earth, describe an area greater for the time ;it must have its orbit less curve, andtherefore approach nearer to the earth in the syzygies than in the quadratures, except in so far as the motion of eccentricity hinders those effects.THE SYSTEM OF THE WORLD. 533Per the eccentricity is greatest when the moon s apogeon is in the syzygies,and least when the same is in the quadratures ; and hence it is that theperigeon moon is swifter and nearer to us, but the apogeon moon slower andfarther from us, in the syzygies than in the quadratures. But farther; theapogeon has a progressive and the nodes a regressive motion, both unequable. For the apogeon is more swiftly progressive in its syzygies, moreslowly regressive in its quadratures, and by the excess of its progress aboveits regress is yearly transferred in coiisequentia ; but the nodes are quiescent in their syzygies, and most swiftly regressive in their quadratures. Butfarther, still, the greatest latitude of the moon is greater in its quadratures than in its syzygies ; and the mean motion swifter in the aphelion ofthe earth than in its perihelion. More inequalities in the moon s motionhave not hitherto been taken notice of by astronomers : but all these follow from our principles in Cor. II, III, IV, V, VI, VII, VIII, IX, X, XI,XII, XIII, Prop. LXVI, and are known really to exist in the heavens.And this may seen in that most ingenious, and if I mistake not, of all, themost acccurate, hypothesis of Mr. Horrnx, which Mr. Flamsted has fittedto the heavens ; but the astronomical hypotheses are to be corrected in themotion of the nodes ;for the nodes admit the greatest equation or prosthaphaeresisin their octants, and this inequality is most conspicuous whenthe moon is in the nodes, and therefore also in the octants ; and hence itwas that Tycho, and others after him, referred this inequality to theoctants of the moon, and made it menstrual; but the reasons by us adduced prove that it ought to be referred to the octants of the nodes, and tobe made annual.Beside those inequalities taken notice of by astronomers (p. 414, 445,447,) there are yet some others, by which the moon s motions are so disturbed, that hitherto by no law could they be reduced to any certain regulation. For the velocities or motions of the apogee and nodes ofthe moon, and their equations, as well as the differs ice betwixt the greatesteccentricity in the syzygies and the least in the < rrdratures, and that inequality which we call the variation, in the progress of the year are augmented and diminished (by Cor. XIV, Prop. LXVI) in the triplicate ratioof the sun s apparent diameter. Beside that, the variation is mutablerly in the duplicate ratio of the time between the quadratures (by Cor.I and II, Lem. X, and Cor. XVI, Prop. LXVI); and all those inequalities are something greater in that part of the orbit which respects the sunthan in the opposite part, but by a difference that is scarcely or not at allperceptible.By a computation (p. 422), which for brevity s sake I do not describe, 1also find that the area which the moon by a radius drawn to the earthdescribes in the several equal moments of time is nearly as the sum of thenumber 237T。, and versed sine of the double distance of the moon frour.534 THE SYSTEM OF THE WORLD.the nearest quadrature in a circle whose radius is unity ; and thereforethat the square of the moon s distance from the earth is as that sum divided by the horary motion of the moon. Thus it is when the variation inthe octants is in its mean quantity ; but if the variation is greater or less,that versed sine must be augmented or diminished in the same ratio. Letastronomers try how exactly the distances thus found will agree with fjiemoon s apparent diameters.From the motions of our moon we may derive the motions of themoon*or satellites of Jupiter and Saturn (p. 413); for the mean motion of thenodes of the outmost satellite of Jupiter is to the mean motion of the nodesof our moon in a proportion compounded of the duplicate proportion ofthe periodic time of the earth about the sun to the periodic time of Jupiterabout the sun, and the simple proportion of the periodic time of the satellite about Jupiter to the periodic time of our moon about the earth (byGor. XVI, Prop. LXVI) : and therefore those nodes, in the space of a hundred years, are carried 8 24 backwards, or in atitecedeutia. The meanmotions of the nodes of the inner satellites are to the (mean) motion of(the nodes of) the outmost as their periodic times to the periodic time ofthis, by the same corollary, and are thence given. And the motion of theapsis of every satellite in consequentia is to the motion of its nodes ina/ttecedentia, as the motion of the apogee of our moon to the motion of i snodes (by the same Corollary), and is thence given. The greatest equations of the nodes and line of the apses of each satellite are to the greatestequations of the nodes and the line of the apses of the moon respectivelyas the motion of the nodes and line of the apses of the satellites in thetime of one resolution of the first equations to the motion of the nodesand apogeon of the moon in the time of one revolution of the last equations. The variation of a satellite seen from Jupiter is to the variationof our moon in the same proportion as the whole motions of their nodesrespectively, during the times in which the satellite and our moon (afterparting from) arc revolved (again) to the sun, by the same Corollary ; amitherefore in the outmost satellite the variation does not exceed 5" 12 ".From the small quantity of those inequalities, and the slowness of themotions, it happens that the motions of the satellites are found to be soregular, that the more modern astronomers either deny all motion to thenodes, or affirm them to be very slowly regressive.(P. 404). While the planets are thus revolved in orbits about remotecentres, in the mean time they make their several rotations about theirproper axes; the sun in 26 days; Jupiter in 9h. 56 ; Mars in 24f,h.;Venus in 23h.; and that in planes not much inclined to the plane of theecliptic, and according to the order of the signs, as astronomers determinefrom the spots or macula? that by turns present themselves to our sight intheir bodies; and there is a like revolution of our earth performed in 24h.;THE SYSTEM OF THE WORLU. 535find those motions are neither accelerated nor retardedl>ythe actions ofthe centripetal forces, as appears by Cor. XXII, Prop. LXVI ; and therefore of all others they are the most equable and most fit for the mensuration of time; but those revolutions are to be reckoned equable not fromtheir return to the sun, but to some fixed star: for as the position of theplanets to the sun is unequably varied, the revolutions of those planetsfrom sun to sun are rendered unequable.In like manner is the moon revolved about its axis by a motion mostequable in respect of the fixed stars, viz., in 27 J. 7h. 43 , that is, in thespace of a sidereal month ;so that this diurnal motion is equal to themean motion of the moon in its orbit : upon which account the same faceof the moon always respects the centre about which this mean motion isperformed, that is, the exterior focus of the moon s orbit nearly ; and hencearises a deflection of the moon s face from the earth, sometimes towardsthe east, and other times towards the west, according to the position of thefocus which it respects ; and this deflection is equal to the equation of themoon s orbit, or to the difference betwixt its mean and true motions; andthis is the moon s libration in longitude: but it is likewise affected witha libration in latitude arising from the inclination of the moon s axis tothe plane of the orbit in which the moon is revolved about the earth;forthat axis retains the same position to the fixed stars nearly, and hence thepoles present themselves to our view by turns, as we may understand fromthe example of the motion of the earth, whose poles, by reason of the inclination of its axis to the plane of the ecliptic, are by turns illuminated bythe sun. To determine exactly the position of the moon s axis to thefixed stars, and the variation of this position, is a problem worthy of anastronomer.By reason of the diurnal revolutions of the planets, the matter whichthey contain endeavours to recede from the axis of this motion ; and hencethe fluid parts rising higher towards the equator than about the poles(p. 405), would lay the solid parts about the equator under water, if thoseparts did not rise also (p. 405, 409) : upon which account the planets aresomething thicker about the equator than about the poles ; and their equinoctial points (p. 413) thence become regressive ; and their axes, by amotion of nutation, twice in every revolution, librate towards their ecliptics, and twice return again to their former inclination, as is explained inCor. XVIII, Prop. LXVI ; and hence it is that Jupiter, viewed throughvery long telescopes, does not appear altogether round (p. 409). but havingits diameter that lies parallel to the ecliptic something longer than thatwhich is drawn from north to south.And from the diurnal motion and the attractions (p. 415, 418) of theBun and moon our sea ought twice to rise and twice to fall every day, aswell lunar as solar (by Cor. XIX, XX, Prop. LXVI), and the greatest636 THE SYSTEM OF THE WORLD.height of the water to happen before the sixth hour of either day and afteithe twelfth hour preceding. By the slowness of the diurnal motion theflood is retracted to the twelfth hour ; and by the force of the motion ofreciprocation it is protracted and deferred till a time nearer to the sixthhour. But till that time is more certainly determined by the phenomena, choosing the middle between those extremes, why may we notconjecture the greatest height of the water to happen at the third hour ?for thus the water will rise all that time in which the force of the luminaries to raise it is greater, and will fall all that time in which their forceis less : viz., from the ninth to the third hour when that force is greater,and from the third to the ninth when it is less. The hours I reckon fromthe appulse of each luminary to the meridian of the place, as well underas above the horizon ; and by the hours of the lunar day I understand thetwenty-fourth parts of that time which the moon spends before it comesabout again by its apparent diurnal motion to the meridian of the placewhich it left the day before.But the two motions which the two luminaries raise will not appear distinguished, but will make a certain mixed motion. In the conjunction or opposition of the luminaries their forces will be conjoined, and bring on thegreatest flood and ebb. In the quadratures the sun will raise the waterswhich the moon dcpresseth. and depress the waters which the moon raiseth;and from the difference of their forces the smallest of all tides will follow.And because (as experience tells us) the force of the moon is greater thanthat of the sun, the greatest height of the water will happen about thethird lunar hour. Out of the syzygies and quadratures the greatest tidewhich by the single force of the moon ought to fall out at the third lunarhour, and by the single force of the sun at the third solar hour, by thecompounded forces of both must fall out in an intermediate time that approaches nearer to the third hour of the moon than to that of the sun:and, therefore, while the moon is passing from the syzygies to the quadratures, during which time the third hour of the sun precedes the third ofthe moon, the greatest tide will precede the third lunar hour, and that bythe greatest interval a little after the octants of the moon ; and by likeintervals the greatest tide will follow the third lunar hour, while the moonis passing from the quadratures to the syzygies.But the effects of the luminaries depend upon their distances from theearth ; for when they are less distant their effects are greater, and whenmore distant their effects are less, and that in the triplicate proportion oftheir apparent diameters. Therefore it is that the sun in the winter time,being then in its perigee, has a greater effect, and makes the tides in thesyzyii ies something greater, and those in the quadratures something less,cre/m.<? panbiis, than in the summer season ; and every month the moon,vhile in the perigee, raiseth greater tides than at the distance of 15 daysTHE SYSTEM OF THE WORLD. 537K NK forc or after, when it is in its apogee. Whence it comes to pasa that twonighest tides do not follow one the other in two immediately succeedingsyzygies.The effect of either luminary doth likewise depend upon its declinationor distance from the equator ;for if the luminary was placed at the pole,it would constantly attract all the parts of the waters, without any intension or remission of its action, and could cause no reciprocation of motion ;and, therefore, as the luminaries decline from the equator towards eitherpole, they will by degrees lose their force, and on this account will excitelesser tides in the solstitial than in the equinoctial syzygies. But in thesolstitial quadratures they will raise greater tides than in the quadraturesabout the equinoxes ; because the effect of the moon, then situated in theequator, most exceeds the effect of the sun ; therefore the greatest tidesfall out in those syzygies. and the least in those quadratures, which happenabout the time of both equinoxes ; and the greatest tide in the syzygies isalways succeeded by the least tide in the quadratures, as we lind by experience. But because the sun is less distant from the earth in winter thanin summer, it cornes to pass that the greatest and least tides more frequently appear before than after the vernal equinox, and more frequentlyafter than before the autumnal.Moreover, the effects cf che luminaries depend upon the latitudes of places.Let AjoEP represent the earth on allsides covered with deep waters: C itscentre; P, p, its poles; AE the equator: P any place without the equator:F/ the parallel of the place : Del thecorrespondent parallel OD the other sideof the equator; L the rlnoe which the moon possessed three hours beforeH the place of the earth directly under it; h the opposite place ; K, k,the places at 90 degrees distance ; CH, Ch, the greatest heights of the seafrom the centre of the earth ; and CK, C&, the least heights : and if withthe axes H/?,, K/r, an ellipsis is described, and by the revolution rf thatellipsis about its longer axis HA a spheroid HPK//jt?A* is formed, this spheroid will nearly represent the figure of the sea; and CF, C/, CD, Cd, willrepresent the sea in the places F,/, D, d. But farther : if in the said revolution of the ellipsis any point N describes the circle NM, cutting theparallels F/, Dr/?in any places R, T, and the equator AE in S, CN willrepresent the height of the sea in all those places R, S, T, situated in thiscircle. Wherefore in the diurnal revolution of any place F the greatestflood will be in F. at the third hour after the appulse of the moon to themeridian above the horizon ; and afterwards the greatest ebb in Q, at thethird hour after the setting of the moon : and then the greatest flood inf.538 THE SYSTEM OF THE WORLD.at the third Lour after the appulse of the rnoon to the meridian under ththorizon , and. lastly, the greatest ebb in Q. at the third hour after therising of the moon; and the latter floodiny" will be less than the preceding flood in F For the whole sea is divided into two huge and hemispherical floods, one in the hemisphere KH/rC on the north side, the otherin the opposite hemisphere KH/cC, whicli we may therefore call the northern and the southern floods : these floods being always opposite the one tothe other, come by turns to the meridians of all places after the intervalof twelve lunar hours ; and, seeing the northern countries partake moreof the northern flood, and the southern countries more of the southernflood, thence arise tides alternately greater and less in all places withoutthe equator in Avhich the luminaries rLe and set. But the greater tidewill happen when the moon declines towards the vertex of the place, aboutthe third hour after the -appulse of the moon to the meridian above thehorizon ; and when the moon changes its declination, that which was thegreater tide will be changed into a lesser; and the greatest difference ofthe floods will fall out about the times of the solstices, especially if theascending node of the moon is about the first of Aries. So the morningtides in winter exceed those of the evening, and the evening tides exceedthose of the morning in summer ; at Plymouth by the height of one foot,but at Bristol by the height of 15 inches, according to the observations ofQvleptvss and Stitrnnj.But the motions which we have been describing suffer some alterationfrom that force of reciprocation which the waters [having once received]retain a little while by their vis iiisita ; whence it comes to pass that thetides may continue for some time, though the actions of the luminariesshould cease. This power of retaining the impressed motion lessens thedifference of the alternate tides, and makes those tides which immediatelysucceed after the syzygies greater, and those which follow next after thequadratures less. And hence it is that the alternate tides at l:1y monthand Bristol do not differ much more one from the other than by the heightof a foot, or of 15 inches; and that the greatest tides <~>f all at those portsare not the first but the third after the syzygies.And. besides, all the motions are retarded in their passage through shallow channels, so that the greatest tides of all, in some strai s and mouthsof rivers, are the fourth, or even the fifth, after the syzygies.It may also happen that the greatest tide may be the fourth or fifthafter the syzygies, or fall out yet later, because the motions of the sea areretarded in passing through shallow places towards the shores: for so thetide arrives at the western coast of Ireland at the third lunar hour, and anhour or two after at the ports in the southern coast of the same island; asalso at the islands Cftssiterides, commonly Sorliti^s ; then successively atPalrnonth. Plymouth, Portland, the isle of Wight, Winchester, Dover,THE SYSTEM OF THE WORLD. 539the mouth of the Thames, arid London Btidgey spending twelve hours inthis passage. But farther; the propagation of the tides may he obstructedeven by the channels of the ocean itself, when they are not of depth enough,for the flood happens at the third lunar hour in the Canary islands ; andat all those western coasts that lie towards the Atlantic ocean, as of Ireland, France, Spain, and all Africa, to the Cape of Good Hope, exceptin some shallow places, where it is impeded, and falls out later; and in thestraits of Gibraltar, where, by reason of a motion propagated from theMediterranean sea, it flows sooner. But, passing from those coasts overthe breadth of the ocean to the coasts of America, the flood arrives first atthe most eastern shores of Brazil, about the fourth or fifth lunar hour;then at the mouth of the river of the Amazons at the sixth hour, but atthe neighbouring islands at the fourth hour ; afterwards at the islands ofBermudas at the seventh hour, and at port St. An^nstin in Florida atseven and a half. And therefore the tide is propagated through the oceanwith a slower motion than it should be according to the course of themoon ; and this retardation is very necessary, that the sea at the same timemay fall between Brazil and New France, and rise at the Canary islands,and on the coasts of Europe and Africa, and vice versa : for the sea cannot rise in one place but by falling in another. And it is probable thatthe Pacific sea is agitated by the same laws : for in the coasts of Chili andPeru the highest flood is said to happen at the third lunar hour. Butwith what velocity it is thence propagated to the eastern coasts ofJapan, the Philippine and other islands adjacent to China, I have notyet learned.Farther; it may happen (p. 418) that the tide may be propagated fromthe ocean through different channels towards the same port, and may passquicker through some channels than through others, in which case thesame tide, divided into two or more succeeding one another, may compoundnew motions of different kinds. Let us suppose one tide to be divided intotwo equal tides, the former whereof precedes the other by the space of sixhours, and happens at the third or twenty-seventh hour from the appulseof the moon to the meridian of the port. If the moon at the time of thisappulse to the meridian was in the equator, every six hours alternately