自然哲学的数学原理-59

there would arise equal floods, which, meeting with as many equal ebbs,would so balance one the other, that, for that day, the water wrould stagnate, and remain quiet. If the moon then declined from the equator, thetides in the ocean would be alternately greater and less, as was said; andfrom hence two greater and two lesser tides would be alternately propagated towards that port. But the two greater floods would make thegreatest height of the waters to fall out in the middle time betwixt both,and the &Greater and lesser floods would make the waters to rise to a meanheight in the middle time between them; and in the middle time between540 THE SYSTEM OF THE WORLD.the two lesser floods the waters would rise to their least height. Thus inthe space of twenty-four hours the waters would come, riot twice, but onceonly to their greatest, and once only to their least height ; and their greatest height, if the moon declined towards the elevated pole, would happenat the sixth or thirtieth hour after the appulse of the moon to the meridianand when the moon changed its declination, this flood would be changedinto an ebb.Of all which we have an example in the port of Batsham, in the kingdom of Tunquin. in the latitude of 20 50 north. In that port, on theday which follows after the passage of the moon over the equator, thewaters stagnate ; when the moon declines to the north, they begin to fluwand ebb, not twice, as in other ports, but once only every day ; and theflood happens at the setting, and the greatest ebb at the rising of the moon.This tide increaseth with the declination of the moon till the seventh oreighth day ; then for the seventh or eighth day following it decreaseth atthe same rate as it had increased before, and ceaseth when the moonchangeth its declination. After which the flood is immediately changedinto an ebb ; and thenceforth the ebb happens at the setting and the floodat the rising of the moon, till the moon again changes its declination.There are two inlets from the ocean to this port; one more direct and shortbetween the island Hainan and the coast of QuanttiHg, a province ofChina ; the other round about between the same island and the coast ofCochim ; and through the shorter passage the tide is sooner propagated toBatsham.In the channels of rivers the influx and reflux depends upon the currentof the rivers, which obstructs the ingress of the waters from the sea. andpromotes their egress to the sea, making the ingress later and slower, andthe egress sooner arid faster; and hence it is that the reflux is of longerduration that the influx, especially far up the rivers, where the force of thesea is less. So Sturmy tells us, that in the river Avon, three miles belowBristol, the water flows only five hours, but ebbs seven ; and without doubtthe difference is yet greater above Bristol, as at Carcs/iam or the Bath.This difference does likewise depend upon the quantity of the flux and reflux;for the more vehement motion of the sea near the syzygies of theluminaries more easily overcoming the resistance of the rivers, will makethe ingress of the water to happen sooner and to continue longer, and willtherefore diminish this difference. But while the moon is approaching tothe syzygies, the rivers will be more plentifully filled, their currents beingobstructed by the greatness of the tides, and therefore will something moreretard the reflux of the sea a little after than a little before the syzygies.Upon which account the slowest tides of all will not happen in the syzy-^ies, but precede them a little; and I observed above that the tides beforethe sy/ygies were also retarded by the force of the sun ; and from bothTHE SYSTEM OF THE WORLD. 541causes conjoined the retardation of the tides will be both greater and soonerbefore the syzygies. All which I find to be so, by the tide-tables whichFtamsted has composed from a great many observations.By the laws we have been describing, the times of the tides are governed ;but the greatness of th-e tides depends upon the greatness of the seas. LetC represent the centre of the earth, EAUB the oval figure of the seas, CAthe longer semi-axis of this oval, OB the shorter insisting at right anglesupon the former, D the middle point between A and B, and EOF or eCfthe angle at the centre of the earth, subtended by the breadth of the seathat terminates in the shores E, F, or e,f. Now, supposing that the pointA is in the middle between the points E, F, and the point D in the middlebetween the points e,/, if the difference of the heights CA, CB, representthe quantity of the tide in a very deep sea surrounding the whole earth,the excess of the height CA above the height OE or OF will represent thequantity of the tide in the middle of the sea EF, terminated by the shoresE, F ; and the excess of the height Ce above the height Cf will nearlyrepresent the quantity of the tide on the shores/" of the same sea. Whenceit appears that the tides are far less in the middle of the sea than at theshores ; and that the tides at the shores are nearly as EF (p. 451, 452), thebreadth of the sea not exceeding a quadrantal arc. And hence it is thatnear the equator, where the sea between Africa and America is narrow,the tides are far less than towards either side in the temperate zones, wrherethe seas are extended wider ; or on almost all the shores of the Pacific sea;as well towards America as towards China,, and within as well as withoutthe tropics ; and that in islands in the middle of the sea they scarcely risehigher than two or three feet, but on the shores of great continents arethree or four times greater, and above, especially if the motions propagatedfrom the ocean are by degrees contracted into a narrow space, and the water,to fill and empty the bays alternately, is forced to flow and ebb with greatviolence through shallow places ; as Plymouth and Chepstow Bridge inEngland) at the mount of >S*/. Michael and town of Avranches in Aormcihdy,and at Cambaia and Peyn. in the East Indies. In which places.642 THE SYSTEM OF THE WORLD.the sea, hurried in and out with great violence, sometimes lays the shoresunder water, sometimes leaves them dry, for many miles. Nor is the forceof the influx and efflux to be broke till it has raised or depressed the waterto forty or fifty feet and more. Thus also -long and shallow straits thatopen to the sea with mouths wider and deeper than the rest of their channel (such as those about Britain and the Magellanic Straits at the eastern entry) will have a greater flood and ebb, or will more intend and remittheir course, and therefore will rise higher and be depressed lower. Orthe coast of South America it is said that the Pacific sea in its refluxsometimes retreats two miles, and gets out of sight of those that stand onshore. Whence in these places the floods will be also higher ; but in deepeiwaters the velocity of influx and efflux is always less, and therefore tluascent and descent is so too. Nor in such places is the ocean known toascend to more than six, eight, or ten feet. The quantity of the ascent Icompute in the following mannerLet S represent the sun, T theearth (419. 420), P the moon,PAGB the moon s orbit. In SPtake SK equal to ST and SL toSK in the duplicate ratio of SKto SP. Parallel to PT draw LM ;and, supposing the mean quantityof the circum-solar force directed towards the earth to be represented 。jthe distance ST or SK, SL will represent the quantity thereof directedtowards the moon. But that force is compounded of the parts SM, LM ;of which the force LM and that part of SM which is represented by TJVI,do disturb the motion of the moon (as appears from Prop. LXVI, and itsCorollaries) In so far as the earth and moon are revolved about theircommon centre of gravity, the earth will be liable to the action of the likeforces. But we may refer the sums as well of the forces as of the motionsto the moon, and represent the sums of the forces by the lines TM andML, which are proportional to them. The force LM, in its mean quantity, is to the force by which the moon may be revolved in an orbit, aboutthe earth quiescent, at the distance PT in the duplicate ratio of the moon speriodic time about the earth to the earth s periodic time about the nun(by Cor. XVII, Prop. LXVI) : that is, in the duplicate ratio of 27d. 7h.43 to 365d. 6h. 9 ;or as 1000 to 178725, or 1 to 178f f. The force bywhich the moon may be revolved in its orb about the earth in rest, at thedistance PT of 60| semi-diameters of the earth, is to the force by whichit may revolve in the same time at the distance of 60 semi- diameters as60i to 60 ; and this force is to the force of gravity with us as 1 to 60 X60 nearly ; and therefore the mean force ML is to the force of gravity atthe surface of the earth as 1 X 60| to 60 X 60 X 178 f, or 1 toTHE SYSTEM OF THF. WORLD. 543638092,6. Whence the force TM will be ulso given from the proportionof the lines TM, Ml,. And these are the forces of the sun, by which themoon s motions are disturbed.If from the moon s orbit (p. 449V we descend to the earth s surface, thoseforces will be diminished in the ratio of the distances 60| and 1 ; andtherefore the force LM will then become 3S604600 times less than theforce of gravity. But this force acting equally every where upon theearth, will scarcely effect any change on the motion of the sea, and therefore may be neglected in the explication of that motion. The other forceI M, in places where the sun is vertical, or in their nadir, is triple thequantity of the force ML, and therefore but 12868200 times less than theforce of gravity.Suppose now AUBE to represent the spherical surface of the enrth,</D/>E the surface of the water overspreading it, C the centre of both, Athe place to winch the sun is vertical, B the place opposite : I), E. placesat 90 degrees distance from the former ; ACEwz/A a right angled cylmdriccanal passing through the earth s centre. The force TM in any place isas the distance of the place from the plane DE, on which a line fr^m Ato C insists at right angles, and J)therefore in the part of the canal which is represented by ECini is of no quantity, but in theother part AClk is as the gravityat the several heights ;for in /descending towards the centre of 7; -pithe earth, gravity is (by Prop- ;LXX1II) every where as theheight ;and therefore the forceTM drawing the water upwardswill diminish its gravity in theleg AC//J of the canal in a givenratio : upon which account thewater will ascend in this leg, till its defect of gravity is supplied by itsgreater height : nor will it rest in an equilibrium till its total gravitybecomes equal to the total gravity in EC/m, the other leg of the canal.Because the gravity of every particle is as its distance from the earth scentre, the weight of the whole water in either leg will increase in theduplicate ratio of the height ; and therefore the height of the water in theleg AC/A* will be to the height thereof in the leg C/wE in the subduplicateratio of the number 12868201 to 12808200, or in the ratio of thenumber 25623053 to the number 25623052, and the height of the waterin the leg EC/ra to the difference of the heights, as 25623052 to 1. Butthe height in the lea: EC/m is of 19615800 Parift feet, as hits been lately544 THE SYSTEM OF THE WORLD.found by the mensuration of the French ; and, therefore, by the precedinganalogy, the difference of the heights comes out 9} inches of the Parisfoot; and the sun s force will make the height of the sea at A to exceedthe height of the same at E by 9 inches. And though the water of thecanal ACE/??7/,: be supposed to be frozen into a hard and solid consistence,yet the heights thereof at A and E, and all other intermediate places, wouldstill remain the same.Let Act (in the following figure) represent that excess of height of 9inches at A, and hf the excess of height at any other place h; and uponDC let fall the perpendicular /G, meeting the globe of the earth in F :and because the distance of the sun ib so great that all the right linesdrawn thereto may be considered as parallel, the force TM in any place /will be to the same force in the place A as the sine FG to the radius AC.And, therefore, since those forces tend to the sun in the direction of parallel lines, they will generatethe parallel heights F/ An,in the same ratio ; and therefore the figure of the waterYlfaeb will be a spheroidmade by the revolution of anellipsis about its longer axisab. And the perpendicularheight fh will be to the oblique height F/ as/G to /C,or as FG to AC : and therefore the height fh is to theheight Art in the duplicateratio of FG to AC, that is, in the ratio of the versed sine of double theangle DC/ to double the radius, and is thence given. And hence to theseveral moments of the apparent revolution of the sun about the earth wemay infer the proportion of the ascent and descent of the waters at anygiven place under the equator, as well as of the diminution of that ascentand descent, whether arising from the latitude of places or from the sun sdeclination ; viz., that on account of the latitude of places,the ascent anddescent of the sea is in all places diminished in the duplicate ratio of theco-sines of latitude ; and on account of the sun s declination, the ascentand descent under the equator is diminished in the duplicate ratio of thev)-sine of declination. And in places without the equator the half sumof the morning and evening ascents (that is, the mean ascent) is diminishednearly in the same ratio.Let S and L respectively represent the forces of the sun and moonplaced in the equator, and at their mean distances from the earth; R theradius ; T and V the versed sines of double the complements of the sunTHE SYSTEM OJ THE WORLD. 545and moon s declinations to any given time ; D and E the moan apparentdiameters of the sun and moon : and, supposing F and G to be their apparent diameters to that given time, their forces to raise the tides under theVG 3 TF 3equator will be, in thesyzygies-^ ^ 1, -f^ 3 S; in the quadratures,VG 3 TF 3--, L -TTT S. And if the same ratio is likewise observed under 2RE 3 2R1) 3the parallels, from observations accurately made in our northern climateswe may determine the proportion of the forces L and S ; and then bymeans of this rule predict the quantities of the tides to every syzygy andquadrature.At the mouth of the river Avon, three miles below Bristol (p. 450 to453), in spring and autumn, the whole ascent of the water in the conjunction or opposition of the luminaries (by the observation of Sturnty) isabout 45 feet, but in the quadratures only 25. Because the apparent diameters of the luminaries are not here determined, let us assume them intheir mean quantities, as well as the moon s declination in the equinoctialquadratures in its mean quantity, that is, 23| ; and the versed sine ofdouble its complement will be 1082, supposing the radius to be 1000. Butthe declinations of the sun in the equinoxes and of the moon in the syzygiesare of no quantity, and the versed sines of double the complementsare each 2000. Whence those forces become L + S in the syzygies, and。 L S in the quadrature^ respectively proportional to the heights/cUOUof the tides of 45 and 25 feet, or of and 5 paces. And, therefore, mul-15138tiplying the extremes and the means, we have 5L + 5S = TxTr L<But farther;I remember to have been told that in summer the ascent ofthe sea in the syzygies is to the ascent thereof in the quadratures as about5 to 4. In the solstices themselves it is probable that the proportion maybe something less, as about 6 to 5 ; whence it would follow that L is =5|S [for then the proportion is L + S : I, -S : : 6 : 5].Till we can more certainly determine the proportion from observation, letus assume L = 5^S ; and since the heights of the tides are as the forceswhich excite them, and the force of the sun is able to raise the tides to theheight of nine inches, the moon s force will be sufficient to raise the sameto the height of four feet. And if we allow that this height may bedoubled, or perhaps tripled, by that force of reciprocation which we observein the motion of the waters, and by which their motion once be ^un is kept35546 THE SYSTEM OF THE WORLD.up for some time, there will be force enough to generate all that quantityof tides which we really find in the ocean.Thus we have seen that these forces are sufficient to move the sea. But.so far as I can observe, they will not be able to produce any other effectsensible on our earth ; for since the weight of one grain in 4000 is notsensible in the nicest balance : and the sun s force to move the tides is12868200 less than the force of gravity ;arid the sum of the forces of bothmoon and sun, exceeding the sun s force only in the ratio of 6^ to 1, is still2032890 times less than the force of gravity ;it is evident that both forcestogether are 500 times less than what is required sensibly to increase * rdiminish the weight of any body in a balance. And, therefore, they willnot sensibly move any suspended body ; nor will they produce any sensibleeifect on pendulums, barometers, bodies swimming in stagnant water, or inthe like statical experiments. In the atmosphere, indeed, they will excitesuch a flux and reflux as they do in the sea, but with so small a motionthat no sensible wind will be thence produced.if the effects of both moon and sun in raising the tides (p. 454), as wellas their apparent diameters, were equal among themselves, their absoluteforces would (by Cor. XIV, Prop. LXVI) be as their magnitudes. But theeffect of the moon is to the effect of the sun as about 5| to 1; and themoon s diameter less than the sun s in the ratio of 31 1 to 32^, or of 45 to46. Now the force of the moon is to be increased in the ratio of the effectdirectly, and in the triplicate ratio of the diameter inversely. Whence theforce of the moon compared with its magnitude will be to the force of thesun compared with its magnitude in the ratio compounded of 5-^- to 1, andthe triplicate of 45 to 46 inversely, that is, in the ratio of about 5^ to 1.And therefore the moon, in respect of the magnitude of its body, has anabsolute centripetal force greater than the sun in respect of the magnitudeof its body in the ratio to 5 T。 to 1, and is therefore more dense in thesame ratio.In the time of 27 1. 7h. 43 , in which the moon makes its revolution aboutthe earth, a planet may be revolved about the sun at the distance of 18.95 1diameters of the sun from the sun s centre, supposing the mean apparendiameter of the sun to be 32} ; and in the same time the moon may be r"-volved about the earth at rest, at the distance of 30 of the earth s diameters. If in both cases the number of diameters was the same, the absolutecircum-terrestrial force would (by Cor. II, Prop. LXXll) be to the absolutecircum-solar force as the magnitude of the earth to the magnitude of thetun. Because the number of the earth s diameters is greater in the ratioof 30 to 18,954, the body of the earth will be less in the triplicate of thatratio, that is, in the ratio of 3|| to 1. Wherefore the earth s force, for themagnitude of its body, is to the sun s force, for the magnitude of its body,as 3f f to 1 : and consequently the earth s density to the sun s will be ILTHE SYSTEM OF THE WORLD 547the same ratio. Since, then, the moon s density is to the sun s density as5JS to I, the moon s density will be to the earth s density as 5 r。 to 3f {,or as 23 to 16. Wh. veforc since the moon s magnitude is to the earth smagnitude as about I to 4l, the moon s absolute centripetal force will beto the earth s absolute centripetal force as about I to 29, and the quantityof matter in the moon to the quantity of matter in the earth in the sameratio.And hence the common centre of gravity of the earth and moon ismore exactly determined than hitherto has been done; from the knowledgeof which AVC may now infer the moon s distance from the earth with greateraccuracy. But I would rather wait till the proportion of the bodies of themoon and earth one to the other is more exactly defined from the phaenomena of the tides, hoping that in the mean time the circumference of theearth may be measured from more distant stations than any body has yetemployed for this purpose.Thus I have given an account of the system of the planets. As to thefixed stars, the smallness of their annual parallax proves them to be removed to immense distances from the system of the planets: that thisparallax is less than one minute is most certain; and from thence it followsthat the distance of the fixed stars is above 360 times greater than thedistance of Saturn from ;he sun. Such as reckon the earth one of theplanets, and the sun one of the fixed stars, may remove the fixed stars toyet greater distances by the following arguments: from the annual motionof the earth there would happen an apparent transposition of the fixedstars, one in respect of another, almost equal to their double parallax: butthe greater and nearer stars, in respect of the more remote, which are onlyseen by the telescope, have not hitherto been observed to have the leastmotion. If we should suppose that motion to be but less than 20", thedistance of the nearer fixed stars would exceed the mean distance of Saturnby above 2000 times. Again: the disk of Saturn, which is only 17" or18" in diameter, receives but about ^---- --^.^ of the sun s light; for somuch less is that disk than the whole spherical surface of the orb of Saturn.Now if we suppose Saturn to rellec* about { of this light, the whole lightreflected from its illuminated hemisphere will be about T ^^Wo o"^~ ^ ^ewhole light emitted from the sun s hemisphere: and, therefore, since lightis rarefied in the duplicate ratio of the distance from the luminous body, ifthe sun was 10000 v/42 times more distant than Saturn, it would yet appear as lucid as Saturn now does without its ring, that is, something morelucid than a fixed star of the first magnitude. Let us, therefore, supposethat the distance from which the sun would shine as a fixed star exceedsthat of Saturn by about 100,000 times, and its apparent diameter will be7V. 16vi. and its parallax arising from the annual motion of the earth 13"" :and so great will be the distance, the apparent diameter, and the parallaxof the fixed stars of the first magnitude, in bulk and light equal to our sun.4>i THE SYSTEM OF THE WORLD.Some may, perhaps, imagine that a great part of the light of the fixed starsis intercepted and lost in its passage through so vast spaces, and upon thataccount pretend to place the fixed stars at nearer distances; but at thisrate the remoter stars could be scarcely seen. Suppose, for example, thatof the light perish in its passage from the nearest fixed stars to us ; then| will twice perish in its passage through a double space, thrice through atriple, and so forth. And, therefore, the fixed stars that are at a doubledistance wHl be 16 times more obscure, viz., 4 times more obscure on account of the diminished apparent diameter ; and, again, 4 times more onaccount of the lost light. And, by the same argument, the fixed stars at atriple distance will be 9 X 4. X 4, or 144 times more obscure; and thoseat a quadruple distance will be 16 X 4 X 4 X 4, or 1024 times more obscure: but so great a diminution of light is no ways consistent with thephenomena and with that hypothesis which places the fixed stars at different distances.Tne fixed stars being, therefore, at such vast distances from one another(p. 460, 461), can neither attract each other sensibly, nor be attracted byour sun. But the comets must unavoidably be acted on by the circumsolar force ; for as the comets were placed by astronomers above the moon,

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