The Tides in North Vietnam

Of all which we have an example in the port of Batsham, in the kingdom of Tunquin (Hai Phong, Vietnam) in the latitude of 20° 50’ north.

In that port, on the day which follows after the passage of the moon over the equator, the waters stagnate.

When the moon declines to the north, they begin to flow and ebb, not twice, as in other ports, but once only every day.

The flood happens at the setting, and the greatest ebb at the rising of the moon.

This tide increases with the declination of the moon till the seventh or eighth day; then for the seventh or eighth day following it decreaseth at the same rate as it had increased before, and ceaseth when the moon changeth its declination.

After which the flood is immediately changed into an ebb; and thenceforth the ebb happens at the setting and the flood at the rising of the moon, till the moon again changes its declination.

There are 2 inlets from the ocean to this port; one more direct and short between the island Hainan and the coast of Quantung, a province of China; the other round about between the same island and the coast of Cochim; and through the shorter passage the tide is sooner propagated to Batsham.

In the channels of rivers the influx and reflux depends upon the current of the rivers, which obstructs the ingress of the waters from the sea, and promotes their egress to the sea, making the ingress later and slower, and the egress sooner and faster; and hence it is that the reflux is of longer duration that the influx, especially far up the rivers, where the force of the sea is less. So Sturmy tells us, that in the river Avon, three miles below Bristol, the water flows only five hours, but ebbs seven; and without doubt the difference is yet greater above Bristol, as at Caresham 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 the luminaries more easily overcoming the resistance of the rivers, will make the ingress of the water to happen sooner and to continue longer, and will therefore diminish this difference. But while the moon is approaching to the syzygies, the rivers will be more plentifully filled, their currents being obstructed by the greatness of the tides, and therefore will something more retard 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 syzygies, but precede them a little; and I observed above that the tides before the syzygies were also retarded by the force of the sun; and from both causes conjoined the retardation of the tides will be both greater and sooner before the syzygies. All which I find to be so, by the tide-tables which Flamsted 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 the tides depends upon the greatness of the seas. Let C represent the centre of the earth, EADB the oval figure of the seas, CA the longer semi-axis of this oval, CB the shorter insisting at right angles upon the former, D the middle point between A and B, and ECF or eCf the angle at the centre of the earth, subtended by the breadth of the sea that terminates in the shores E, F, or e, f.

Supposing that the point A is in the middle between the points E, F, and the point D in the middle between the points e, f, if the difference of the heights CA, CB, represent the quantity of the tide in a very deep sea surrounding the whole earth, the excess of the height CA above the height CE or CF will represent the quantity of the tide in the middle of the sea EF, terminated by the shores E, F

The excess of the height Ce above the height Cf will nearly represent the quantity of the tide on the shores f of the same sea. Whence it appears that the tides are far less in the middle of the sea than at the shores; and that the tides at the shores are nearly as EF (p. 451, 452), the breadth of the sea not exceeding a quadrantal arc. And hence it is that near the equator, where the sea between Africa and America is narrow, the tides are far less than towards either side in the temperate zones, where the 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 without the tropics; and that in islands in the middle of the sea they scarcely rise higher than two or three feet, but on the shores of great continents are three or four times greater, and above, especially if the motions propagated from 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 great violence through shallow places; as Plymouth and Chepstow Bridge in England, at the mount of St. Michael and town of Avranches in Normandy, and at Cambaia and Pegu in the East Indies.

In which places, the sea, hurried in and out with great violence, sometimes lays the shores under water, sometimes leaves them dry, for many miles. Nor is the force of the influx and efflux to be broke till it has raised or depressed the water to forty or fifty feet and more. Thus also long and shallow straits that open 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 remit their course, and therefore will rise higher and be depressed lower. Or the coast of South America it is said that the Pacific sea in its reflux sometimes retreats two miles, and gets out of sight of those that stand on shore. Whence in these places the floods will be also higher but in deeper waters the velocity of influx and efflux is always less, and therefore the ascent and descent is so too. Nor in such places is the ocean known to ascend to more than six, eight, or ten feet. The quantity of the ascent I compute in the following manner.

Let S represent the sun, T the earth (419, 420), P the moon, PAGB the moon’s orbit. In SP take SK equal to ST and SL to SK in the duplicate ratio of SK to SP. Parallel to PT draw LM; and, supposing the mean quantity of the circum-solar force directed towards the earth to be represented by the distance ST or SK, SL will represent the quantity thereof directed towards 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 TM, do disturb the motion of the moon (as appears from Prop. LXVI, and its Corollaries). In so far as the earth and moon are revolved about their common centre of gravity, the earth will be liable to the action of the like forces. But we may refer the sums as well of the forces as of the motions to the moon, and represent the sums of the forces by the lines TM and ML, 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, about the earth quiescent, at the distance PT in the duplicate ratio of the moon’s periodic time about the earth to the earth’s periodic time about the sun (by Cor. XVII, Prop. LXVI); that is, in the duplicate ratio of 27d.7h.43′ to 365d.6d.9′; or as 1000 to 178725, or 1 to 17829⁄40. The force by which the moon may be revolved in its orb about the earth in rest, at the distance PT of 60½ semi-diameters of the earth, is to the force by which it may revolve in the same time at the distance of 60 semi-diameters as 60½ to 60; and this force is to the force of gravity with us as 1 to 60 × {\displaystyle \scriptstyle \times } 60 nearly; and therefore the mean force ML is to the force of gravity at the surface of the earth as 1 × {\displaystyle \scriptstyle \times } 60½ to 60 × {\displaystyle \scriptstyle \times } 60 × {\displaystyle \scriptstyle \times } 17829⁄40, or 1 to 638092,6. Whence the force TM will be also given from the proportion of the lines TM, ML. And these are the forces of the sun, by which the moon’s motions are disturbed.

If from the moon’s orbit (p. 449) we descend to the earth’s surface, those forces will be diminished in the ratio of the distances 60½ and 1; and therefore the force LM will then become 38604600 times less than the force of gravity. But this force acting equally every where upon the earth, will scarcely effect any change on the motion of the sea, and therefore may be neglected in the explication of that motion. The other force TM, in places where the sun is vertical, or in their nadir, is triple the quantity of the force ML, and therefore but 12868200 times less than the force of gravity.

Suppose now ADBE to represent the spherical surface of the earth, aDbE the surface of the water overspreading it, C the centre of both, A the place to winch the sun is vertical, B the place opposite; D, E, places at 90 degrees distance from the former; ACEmlk a right angled cylindric canal passing through the earth’s centre. The force TM in any place is as the distance of the place from the plane DE, on which a line from A to C insists at right angles, and therefore in the part of the canal which is represented by EClm is of no quantity, but in the other part AClk is as the gravity at the several heights; for in descending towards the centre of the earth, gravity is (by Prop LXXIII) every where as the height; and therefore the force TM drawing the water upwards will diminish its gravity in the leg AClk of the canal in a given ratio: upon which account the water will ascend in this leg, till its defect of gravity is supplied by its greater height; nor will it rest in an equilibrium till its total gravity becomes equal to the total gravity in EClm, the other leg of the canal. Because the gravity of every particle is as its distance from the earth’s centre, the weight of the whole water in either leg will increase in the duplicate ratio of the height; and therefore the height of the water in the leg AClk will be to the height thereof in the leg ClmE in the subduplicate ratio of the number 12868201 to 12868200, or in the ratio of the number 25623053 to the number 25623052, and the height of the water in the leg EClm to the difference of the heights, as 25623052 to 1. But the height in the leg EClm is of 19615800 Paris feet, as has been lately found by the mensuration of the French; and, therefore, by the preceding analogy, the difference of the heights comes out 91⁄5 inches of the Paris foot; and the sun’s force will make the height of the sea at A to exceed the height of the same at E by 9 inches. And though the water of the canal ACEmlk be supposed to be frozen into a hard and solid consistence, yet the heights thereof at A and E, and all other intermediate places, would still remain the same.

Let Aa (in the following figure) represent that excess of height of 9 inches at A, and hf the excess of height at any other place h; and upon DC let fall the perpendicular fG, meeting the globe of the earth in F; and because the distance of the sun is so great that all the right lines drawn thereto may be considered as parallel, the force TM in any place f 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 generate the parallel heights Ff, Aa, in the same ratio; and therefore the figure of the water Dfaeb will be a spheroid made by the revolution of an ellipsis about its longer axis ab. And the perpendicular height fh will be to the oblique height Ff as fG to fC, or as FG to AC: and therefore the height fh is to the height Aa in the duplicate ratio of FG to AC, that is, in the ratio of the versed sine of double the angle DCf to double the radius, and is thence given. And hence to the several moments of the apparent revolution of the sun about the earth we may infer the proportion of the ascent and descent of the waters at any given place under the equator, as well as of the diminution of that ascent and descent, whether arising from the latitude of places or from the sun’s declination; viz., that on account of the latitude of places, the ascent and descent of the sea is in all places diminished in the duplicate ratio of the co-sines of latitude; and on account of the sun’s declination, the ascent and descent under the equator is diminished in the duplicate ratio of the co-sine of declination. And in places without the equator the half sum of the morning and evening ascents (that is, the mean ascent) is diminished nearly in the same ratio.

Let S and L respectively represent the forces of the sun and moon placed in the equator, and at their mean distances from the earth; R the radius; T and V the versed sines of double the complements of the sun and moon’s declinations to any given time; D and E the mean apparent diameters 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 the equator will be, in the syzygies … in the quadratures,

If the same ratio is likewise observed under the parallels, from observations accurately made in our northern climates we may determine the proportion of the forces L and S; and then by means of this rule predict the quantities of the tides to every syzygy and quadrature.

At the mouth of the river Avon, three miles below Bristol (p. 450 to 453), in spring and autumn, the whole ascent of the water in the conjunction or opposition of the luminaries (by the observation of Sturmy) is about 45 feet, but in the quadratures only 25. Because the apparent diameters of the luminaries are not here determined, let us assume them in their mean quantities, as well as the moon’s declination in the equinoctial quadratures in its mean quantity, that is, 23½°; and the versed sine of double its complement will be 1682, supposing the radius to be 1000. But the declinations of the sun in the equinoxes and of the moon in the syzygies are of no quantity, and the versed sines of double the complements are each 2000. Whence those forces become L + S in the syzygies, and 1682⁄2000 L - S in the quadratures; respectively proportional to the heights of the tides of 45 and 25 feet, or of 9 and 5 paces. And, therefore, multiplying the extremes and the means, we have 5L + 5S = 15138⁄2000L - 9S, or L = 28000⁄5138S = 55⁄11S.

But farther; I remember to have been told that in summer the ascent of the sea in the syzygies is to the ascent thereof in the quadratures as about 5 to 4. In the solstices themselves it is probable that the proportion may be something less, as about 6 to 5; whence it would follow that L is 51⁄6S [for then the proportion is 1682 2000 L + 1682 2000 S : L − 1682 2000 S :: 6 : 5 {\displaystyle \scriptstyle {\frac {1682}{2000}}L+{\frac {1682}{2000}}S:L-{\frac {1682}{2000}}S::6:5}]. Till we can more certainly determine the proportion from observation, let us assume L = 5⅓S; and since the heights of the tides are as the forces which excite them, and the force of the sun is able to raise the tides to the height of nine inches, the moon’s force will be sufficient to raise the same to the height of four feet. And if we allow that this height may be doubled, or perhaps tripled, by that force of reciprocation which we observe in the motion of the waters, and by which their motion once begun is kept up for some time, there will be force enough to generate all that quantity of 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 effect sensible on our earth; for since the weight of one grain in 4000 is not sensible in the nicest balance; and the sun’s force to move the tides is 12868200 less than the force of gravity; and the sum of the forces of both moon and sun, exceeding the sun’s force only in the ratio of 6⅓ to 1, is still 2032890 times less than the force of gravity; it is evident that both forces together are 500 times less than what is required sensibly to increase or diminish the weight of any body in a balance. And, therefore, they will not sensibly move any suspended body; nor will they produce any sensible effect on pendulums, barometers, bodies swimming in stagnant water, or in the like statical experiments. In the atmosphere, indeed, they will excite such a flux and reflux as they do in the sea, but with so small a motion that no sensible wind will be thence produced.

If the effects of both moon and sun in raising the tides (p. 454), as well as their apparent diameters, were equal among themselves, their absolute forces would (by Cor. XIV, Prop. LXVI) be as their magnitudes. But the effect of the moon is to the effect of the sun as about 5⅓ to 1; and the moon’s diameter less than the sun’s in the ratio of 31½ to 32⅓, or of 45 to 46. Now the force of the moon is to be increased in the ratio of the effect directly, and in the triplicate ratio of the diameter inversely. Whence the force of the moon compared with its magnitude will be to the force of the sun compared with its magnitude in the ratio compounded of 5⅓ to 1, and the triplicate of 45 to 46 inversely, that is, in the ratio of about 57⁄10 to 1. And therefore the moon, in respect of the magnitude of its body, has an absolute centripetal force greater than the sun in respect of the magnitude of its body in the ratio to 57⁄10 to 1, and is therefore more dense in the same ratio.

In the time of 27d.7h.43’, in which the moon makes its revolution about the earth, a planet may be revolved about the sun at the distance of 18,954 diameters of the sun from the sun’s centre, supposing the mean apparent diameter of the sun to be 321⁄5’; and in the same time the moon may be revolved 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 absolute circum-terrestrial force would (by Cor. II, Prop. LXXII) be to the absolute circum-solar force as the magnitude of the earth to the magnitude of the sun. Because the number of the earth’s diameters is greater in the ratio of 30 to 18,954, the body of the earth will be less in the triplicate of that ratio, that is, in the ratio of 328⁄29 to 1. Wherefore the earth’s force, for the magnitude of its body, is to the sun’s force, for the magnitude of its body, as 328⁄29 to 1; and consequently the earth’s density to the sun’s will be in the same ratio. Since, then, the moon’s density is to the sun’s density as 57⁄10 to 1, the moon’s density will be to the earth’s density as 57⁄10 to 328⁄29, or as 23 to 16. Wherefore since the moon’s magnitude is to the earth’s magnitude as about 1 to 41½, the moon’s absolute centripetal force will be to the earth’s absolute centripetal force as about 1 to 29, and the quantity of matter in the moon to the quantity of matter in the earth in the same ratio. And hence the common centre of gravity of the earth and moon is more exactly determined than hitherto has been done; from the knowledge of which we may now infer the moon’s distance from the earth with greater accuracy. But I would rather wait till the proportion of the bodies of the moon and earth one to the other is more exactly defined from the phænomena of the tides, hoping that in the mean time the circumference of the earth may be measured from more distant stations than any body has yet employed for this purpose.

Thus I have given an account of the system of the planets. As to the fixed stars, the smallness of their annual parallax proves them to be removed to immense distances from the system of the planets: that this parallax is less than one minute is most certain; and from thence it follows that the distance of the fixed stars is above 360 times greater than the distance of Saturn from the sun. Such as reckon the earth one of the planets, and the sun one of the fixed stars, may remove the fixed stars to yet greater distances by the following arguments: from the annual motion of the earth there would happen an apparent transposition of the fixed stars, one in respect of another, almost equal to their double parallax; but the greater and nearer stars, in respect of the more remote, which are only seen by the telescope, have not hitherto been observed to have the least motion. If we should suppose that motion to be but less than 20", the distance of the nearer fixed stars would exceed the mean distance of Saturn by above 2000 times. Again; the disk of Saturn, which is only 17" or 18" in diameter, receives but about 1⁄2100000000 the sun’s light; for so much less is that disk than the whole spherical surface of the orb of Saturn. Now if we suppose Saturn to reflect about ¼ of this light, the whole light reflected from its illuminated hemisphere will be about 1⁄4200000000 of the whole light emitted from the sun’s hemisphere; and, therefore, since light is rarefied in the duplicate ratio of the distance from the luminous body, if the sun was 10000 42 {\displaystyle \scriptstyle {\sqrt {42}}} times more distant than Saturn, it would yet appear as lucid as Saturn now does without its ring, that is, something more lucid than a fixed star of the first magnitude. Let us, therefore, suppose that the distance from which the sun would shine as a fixed star exceeds that of Saturn by about 100,000 times, and its apparent diameter will be 7v.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 parallax of the fixed stars of the first magnitude, in bulk and light equal to our sun. Some may, perhaps, imagine that a great part of the light of the fixed stars is intercepted and lost in its passage through so vast spaces, and upon that account pretend to place the fixed stars at nearer distances; but at this rate the remoter stars could be scarcely seen. Suppose, for example, that ¾ of the light perish in its passage from the nearest fixed stars to us; then ¾ will twice perish in its passage through a double space, thrice through a triple, and so forth. And, therefore, the fixed stars that are at a double distance will be 16 times more obscure, viz., 4 times more obscure on account of the diminished apparent diameter; and, again, 4 times more on account of the lost light. And, by the same argument, the fixed stars at a triple distance will be 9 × {\displaystyle \scriptstyle \times } 4 × {\displaystyle \scriptstyle \times } 4, or 144 times more obscure; and those at a quadruple distance will be 16 × {\displaystyle \scriptstyle \times } 4 × {\displaystyle \scriptstyle \times } 4 × {\displaystyle \scriptstyle \times } 4, or 1024 times more obscure: but so great a diminution of light is no ways consistent with the phænomena and with that hypothesis which places the fixed stars at different distances.