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of high and low Water the Quantities sy1-ss and tv-tt are always in the constant Ratio of 29 L to 30 S, or of 20 X5 to 30X2; but the Quantity ——SS can never exceed; confequently Vi—tt can

never

exceed.

3XI
29× 5

or; and of course one of the Factors t or VI-tt must be always very fmall, which proves that the Moon is near the Meridian at High-Water, and near the Horizon at Low-Water.

The waters The Waters of the Sea therefore fhould be elevated and depreffed of the Sea twice in the Space of a lunar Day, that is in the Interval of Time ought twice to rife and elapfed between the Paffage of the Moon at the Meridian of any Place, twice to fall and its Return to the fame Meridian; for the conjoint Force of the every day. Sun and Moon on the Sea, being greatest when the Moon is near the Meridian, it fhould be equal twice in 24 Hours 49 Minutes (a), when the Moon is near the Meridian of the Place above and below the Horizon; wherefore in each diurnal Revolution of the Moon about the Earth, there should be two Tides diftant from each other, by the fame Interval that the Moon employs to pafs from the Meridian above the Horizon to that below it, which Interval is about 12h 24m. hence the Time of High-Water will be later and later every Day.

High water does not im

mediately follow the Appulfe of

the Moon

to the Meridian.

XVI.

Since tv-tt can never exceed, and confequently the Distance of the Moon from the Meridian 12 Degrees, the greatest Elevation of the Waters in any Place can never happen later than 48 lunar Minutes, or 50 folar Minutes after the Appulfe of the Moon to the Meridian, if the Waters had no Inertia, and their Motion were not retarded by their Friction again the Bottom of the Sea. But from thofe two Caufes this Elevation ftill happens two Hours and a Half or three Hours later

(a) Whilft the Heavens feem to carry the Sun and Moon round from Eaft to West every Day, thofe Luminaries move in a contrary Direction, the Sun 59 m. 85.,3 the Moon 13 d. 1cm. 358. in a Day, confequently after their Conjunction the Moon continually recedes 12 d. 11m. 265. 7 from the Sun towards the Eaft each Day, until she is 130 Degrees from the Sun, or in Oppofition, after which being to the Weft of the Sun, the continually approaches, and at length overtakes him in 29 Days and an Half. From whence it appears that this Planet, the Day of the new Moon, rifes, paffes at the Meridian and fets about the fame Time as the Sun; the following Days the rifes, paffes at the Meridian, and fets later and later than the Sun, fo that the mean Quantity of the Retardation of one rifing compared with the following, of one Appulfe to the Meridian compared with the following, &c. is about 49 Minutes. Seven Days and One-third after the Conjunction, the Moon being 90 Degrees to the Eaft of the Sun, or in its first Quarter, the rifes when the Sun is in the Meridian, paffes at the Meridian when the Sun fets, and fets at Midnight. The following Days the comes fooner to the Meridian than the Sun to the oppofite Meridian, but the Difference continually decreases to the Oppofition, and then the rifes when the Sun fets, paffes at the Meridian at Midnight, and fets when the Sun rifes. The following Days fhe comes later and later to the Meridian than the Sun to the oppofite Meridian, the Dif ference increafing to the laft Quarter when the Moon being 90 Degrees to the Weft of the Sun, rifes at Midnight, pafles at the Meridian at Six of the Clock in the Morning and fets at Noon. The following Day's the rifes, pafles at the Meridian, and sets sooner than the Sun, the Interval decreafing to the Conjunction.

in the Ports of the Ocean where the Sea is open; for the Waters in confequence of their Force of Inertia receiving but by Degrees their Motion, and retaining for fome Time the Motion they have acquired, the Motion of the Sea is perpetually accelerated during the fix Hours which precedes the Appulfe of the Moon to the Meridian, by the combined Actions of the Sun and Moon on the Waters, which increases in proportion as the Moon rifes above the Horizon, and by the diurnal Motion of the Earth which then confpires with that of the Moon. This Mo- What are tion impressed on the Waters retains during fome Time its Acceleration, the Caufes fo that the Sea rises higher and higher until the diurnal Motion of the Earth which becomes contrary after the Appulfe of the Moon to the Meridian, as alfo the combined Actions of the Luminaries which becomes weaker and weaker, diminishes gradually the Velocity of the Waters, in confequence of which they fink. It is eafy to perceive that the Friction of the Waters against the Bottom of the Sea fhould also contribute to retard the Tides.

In the Regions where the Sea has no Communication with the Ocean, the Tides are much more retarded, in fome Places even 12 Hours, and it is ufual to say in thofe Places, that the Tides precede the Appulfe of the Moon to the Meridian. In the Port of Havre-de-grace, for Example, where the Tide retards 9 Hours, it is imagined that it precedes by 3 Hours the Appulfe of the Moon to the Meridian; but in Reality, this Tide is the Effect of the precedent Culmination.

which retard

the Tides.

between the

The Waters falling to the lowest when the Moon is near the Horizon, Low-water her Action on the Sea being then most oblique, it is manifest that Low- does not water does not exactly fall between the two High-waters which immedi- exactly fall ately fucceed each other, but is so much nearer to that which follows, as two Elevathe Elevation of the Pole in the propofed Place is greater, and the Moon tions which immediately has a greater Declination; that is, in proportion to the Interval between fucceed the rifing and fetting of the Moon and the horary Circle of fix Hours each other, and why. after her Culmination.

XVII.

ftrual Variations.

These are the principal Phenomena which attend the Tides depend- The mening on the Position of the different Parts of the Earth in its diurnal Revolution with respect to the Sun and Moon. We shall now proceed to explain the Variations in the Tides which happen every Month, and which depend on the Change of Pofition of the Moon with Refpect to the Sun and the Earth.

XVIII.

In the Conjunction of the Sun and Moon, thofe Luminaries coming The greatto the Meridian at the fame Time, and in the Oppofition when one eft Tides comes to the Meridian the other coming to the oppofite Meridian, they happen and confpire to raise the Waters of the Sea. In the Quadratures on the full Moon.

at

tures.

The leaft in contrary the Waters raised by the Sun, are depreffed by the Moon, the the Quadra- Waters under the Moon being 90 Degrees from thofe under the Sun; confequently the greatest Tides happen at full and new Moon, and the leaft at first and last Quarter.

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The great

on of the

Meridian

whilft the

dratures,

XIX.

The greatest and leaft Tides do not happen in the Syfigies and Quadratures, but are the Third or the Fourth in Order after the Syfigies and Quadratures, becaufe as in other Cafes fo in this, the Effect is not the greatest or the leaft when the immediate Influence of the Caufe is greatest or leaft. If the Sea was perfectly at Reft when the Sun and Moon act on it in the Syfigies, it would not inftantly attain its greateft Velocity, nor confequently its greatest Height, but would acquire it by Degrees. Now as the Tides which precede the Syfigies are not the greateft, they increase gradually, and the Waters have not acquired their greatest Height until fome Time after the Moon has paffed the Syfigies, and he begins to counteract the Sun's Force and depress the Waters where the Sun raifes them. Likewife the Tides which precede the Quadratures are not the leaft, they decrease gradually and do not come to their leaft Height until fome Time after the Moon has paffed the Quadratures.

XX.

The greatest Height of the Waters which by the fingle Force of the eft Elevati- Moon would happen at the Moon's Appulfe to the Meridian, and by Waters hap- the fingle Force of the Sun at the Sun's Appulfe to the Meridian, abpens focner ftracting from the external Caufes which retard it; by the combined after the Ap pulfe of the Forces of both muft fall out in an intermediate Time, which correfMcon to the ponds more exactly to the Motion of the Moon than to that of the Sun, wherefore when the Moon paffes from Conjunction or Oppofition to pailes from Quadrature, this greatest Height answers more to the fetting of the the Syfigies Moon. The Sun in the first Cafe coming fooner to the Meridian than to the Qua- the Moon, and in the latter the Moon coming later to the Meridian and later than the Sun to the oppofite Meridian; and when the Moon paffes whilft the from Quadrature to Oppofition or Conjunction, this greatest Elevation Moon paffes from the anfwers more to the rifing of the Moon. In the first Cafe, the Moon Quadratures coming fooner to the Meridian than the Sun to the oppofite Meridian, and in the latter, the Moon coming fooner to the Meridian than the Sun (b). To calculate thofe Variations in the Time of High-water which arife from the refpective Pofitions of the Sun and Moon, let us fuppofe on a certain Day, the Sun and Moon to be in Conjunction at the Appulfe of the Moon to the Meridian of any Place, and confequently that it is High-Water there at that Inftant. The following Day at the

to the

Syfigies.

(b) See preceding Note

Time of High-Water in faid Place, the Sum of the Distances (z'x') of the Sun and Moon from the Meridian will be 124. 30m, and the Interval between the two Tides will be expreffed in folar Hours by 360d.+Arc z'. Since the Arcs z' and x' are very fmall, they may be fuppofed without any fenfible Error to coincide with their Sines (VI—SS) (VI-t) and VI-ss+VI-tt may be fuppofed equal to Sin. 124. 30m, =0,21643, and confequently VI-tt=0, 21643-VI-ss, we may fuppose alfo s=1 and 11: after thofe Subftitutions the Equation

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2

L
S

tuting for we will have

VI-55
0,21643-VI-55
V1-55
0,21643-VISS 12

29 L
X.

[ocr errors]

22 x; and fubfti

30

29 which gives for

VI-ss or for the Sine of the Arc z' required 29 X0,21643=0, 15308 or z'-84. 48m, or 353 folar Minutes, fo that the whole Interval is 24h. 35m..

Let us now fuppofe on a certain Day, the Sun and Moon to be in Quadrature at the Appulfe of the Moon to the Meridian at the above mentioned Place, and confequently that it is High-Water there at that Inftant; the following Day at the Time of High-water the Sum of the Distances (z+x') of the Sun and Moon from the Meridian (if it be the laft Quadrature) will be 77 Degrees, and the Sum of the Distances (z+z') of the Sun from the Horizon and Meridian being 90 Degrees, z-x'=12d. 30m, that is, s-VI-tt=0, 21643 and VI-tt-s-0, 21643. But in this Cafe v1-s may be fuppofed =1 and 1, wherefore 22 which gives s0,36920 answer

[ocr errors]

tVI―tt

=

[ocr errors]

S 21643

[ocr errors]

ing to 21d. 4om, or to th 252 Minutes, fo that the whole Interval (360d.+Arc z) is 25 Hours, 26 Minutes.

From whence it appears the High-Water fhould precede the Appulfe of the Moon to the Meridian whilst the is paffing from the Syfigies to the Quadratures, and fhound .llow the Abulfe of the Moon to the Meridian whilft the is paffing from the Quadratures to the Syfigies; that the greatest Anticipation Retardation fhould be about 50 folar Minutes, and that the Distance of the Sun and Moon from each other at the Time of the greatest Anticipation or Retardation is about 57 Degrees. But from external Cafes High-Water happens in the Ports of the Ocean three Hours later, codequently in thofe Ports it fhould precede the third lunar Hour, and that by the greatest Interval the ninth Tide after the Syfigies, and this great eft Anticipation being repaired in the five fubfequent Tides, it fhould follow by like Intervals the third lunar Hour, whilst the Moon is paffing from the Quadratures to the Syfigies.

The Tides

are greater ceteris paribus, when the Moon

is in Perigee than when

she is in Apogee. The anual Variations, the Tides are greater

in Winter than in Summer.

The Tides

depend on tion of the

the Declina

Sun and
Moon.

The Time

XXI.

Finally, all other Circumstances being alike, the Tides are greatest in the fame Aspects of the Sun and Moon, when they have the fame De. clination, when the Moon is in Perigee than when the is in Apogee. The Force of the Moon on the Waters of the Sea decreasing in the triplicate Ratio of her Distance from the Earth.

XXII.

The annual Variations of the Tides depend on the Distance of the Earth from the Sun, hence it is that in Winter the Tides are greater, all other Circumstances being alike, in the Syfigies, and less in the Quadratures than in Summer, the Sun being nearer to the Earth in Winter than in Summer.

XXIII.

The Effects of the Sun and Moon upon the Waters of the Sea depend upon the Declination of the Luminaries, for if either the Sun or Moon was in the Pole, any Place of the Earth in describing its Parallel to the Equator, would not meet in its Course with any Part of the Water more elevated than another, so that there would be no Tide in any Place; therefore the Actions of the Sun and Moon on the Waters of the Sea become weaker as they decline from the Equator, and Newton found (Prop. 37. B. 3.) that the Force of each Luminary on the Sea decreases in the duplicate Ratio of the Cofine of its Declination; hence it is, that the Tides in the folfticial Syfigies are lefs than in the equinoctial Syfigies, and are greater in the folfticial Quadratures than in the equinoctial Quadratures, because in the folfticial Quadratures the Moon is in the Equator, and in the other the Moon is in one of the Tropics, and the Tide depends more on the Action of the Moon than that of the Sun, and is therefore greatest when the Moon's Action is greatest.

The Spring Tides therefore ought to be the greateft, and the Neap Tides the leaft at the Equinoxes, and because the Sun is nearer the Earth in Winter than in Summer, the Spring Tides are greatest and the Neap Tides the least in Winter; hence it is, that the greatest Spring and leaft Neap Tides are after the autumnal and before the vernal Equinox.

Two great Spring Tides never follow each other in the two nearest Syfigies, becaufe if the Moon in one of the Syfigies be in her Perigee, fhe will in the following Syfigie be in her Apogee. In the first Cafe her Action being greatest and confpiring with that of the Sun, the Waters will be raised to their greatest Height; but in the latter Case her Action being leaft, the Tide will be lefs.

XXIV.

The ebbing and flowing of the Sea depends alfo upon the Latitude of and Height the Place; for the conjoint Actions of the Sun and Moon changing the depend up- Water upon the Earth's Surface into an oblong Spheroid, one of the

of the Tides

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