Computation founded on the Laws of Equilibrium, wherefore the great Spring Tides and Neap Tides are in a greater Ratio according to the Laws of Equilibrium than that of 9 to 5.

Bernoully fuppofes them to be to each other as 7 to 3, confequently that Force of the Force (L) of the Moon is to the Force (S) of the Sun as 5 to 2. A proaccording to portion which answers better to the Obferved Variations in the duratiBernoully. on and interval of the Tides (Variations which receive no Alteration from

the moon

Singular

moon.

the above mentioned fecondary Caufes) and to the other Theories which depend on a Determination of the Force of the Moon. Hence the Density of the Moon is to the Denfity of the Earth as 7 to 10, the Quantity of Matter in the Moon is to the Quantity of Matter in the Earth as I to 70, and finally the accelerative Gravity at the Surface of the Moon is to the accelerative Gravity on the Surface of the Earth as 1 to 5.

XIV.

If the Moon's Body were Fluid like our Sea it would be elevated by the figure of the Action of the Earth in the Parts which are nearest to it and in the Parts oppofite to these, and Newton enquires into the Quantity of this Elevation. He obferves that the Elevation (8) of the Earth produced by the Action of the Moon would be to the Elevation (E) of the Moon (if it had the fame Diameter as the Earth) produced by the Action of the Earth as the Quantity of Matter in the Moon to the Quantity of Matter in the Earth, or as 1 to 39,788. and the Elevation (E) produced by the Action of the Earth in the Moon if it had the fame Diameter as the Earth, is to the real Elevation (x) produced in the Moon by the Action of the Earth, as the Diameter of the Earth to the Diameter of the Moon or as 365 to 100. wherefore by the Compofition of Ratios 8XE is to Exx or the Elevation of the Earth (83) produced by the Action of the Moon is to the real Elevation of the Moon produced by the Action of the Earth as 1X 365 to 39,788 X 100 or as 1081 to 100 or x=93 Feet, confequently the Diameter of the Moon that paffes through the Centre of the Earth, muft exceed the Diameter which is perpendicular to it by 186 Feet. Hence it is, that the Moon always turns the fame Side towards the Earth.

1pheroidal

117000.

In Effect La Grange in his Piece which carried the Prize of the royal Effet of Academy of Sciences in the Year 1764, fuppofing with Newton that the the obling Moon is a Spheroid having its longeft Diameter directed towards the Earth, figure of the has found that this Planet fhould have a libratory or ofcillatory Motion about its Axis, whereby its Velocity of Rotation is fometimes accelerated and other Times retarded, and that then the Moon fhould always turn the fame Bide nearly towards the Earth, though it did not receive in the Beginning a Motion of Rotation whofe Duration was equal to that of its Revolution. La Grange has demonftrated alfo that the Figure of the Moon might be fuch that the Preceffion of its equinoctial Points or the Retrogradation of the

Nodes of the lunar Equator, would be equal to the retrograde Motion of the Nodes of the lunar Orbit; and in this Cafe he found that the lunar Axis would have no fenfible Nutation. The Action of the Sun in all those Inquiries, is almost infenfible with refpect to that of the Earth; it is the Earth which produces the Motion of the Nodes of the lunar Equator, by acting more or less obliquely on the lunar Spheroid; hence the Preceffion of the lunar Equator, and the Law of the Motion produced in the lunar Spheroid, differ very much from that which is obferved in the Equator of the Earth. The Researches of this eminent Mathematician of Turin, fhall be explained hereafter.

XV.

Newton having fhewn that the Tides proceed from the combined Actions of the Sun and Moon, and determined the Quantity that each of those Luminaries contribute to their Production, enters into an Explanation of the Circumstances which attend the Phenomena of the Tides.

ons have

There has been observed in all Times, three Kinds of Motions in the Three kinds Sea, its diurnal Motion, whereby it ebbs and flows twice a Day, the of variatiregular Alterations which this Motion receives every Month, and which been obfollow the Position of the Moon with refpect to the Sun, and those served in which arrive every Year and which depend on the Pofition of the Earth the motion with refpect to the Sun.

of the fea.

To deduce thofe Motions from their Cause, we are to obferve that Diurnal the Sea yielding to the Force of the Sun and Moon impreffed on it in variations. Proportion to their Quantity, acquires its greatest Height by a Force compounded of those two Forces; hence this greatest Height (even abftracting from the Force of Inertia of the Waters) should not be immediately under the Moon, nor immediately under the Sun, but in an intermediate Point, which corresponds more exactly to the Motion of the Moon than to that of the Sun, because the Force of the Moon on the Sea is greater than that of the Sun. To determine the Pofition of this Point, it is manifest that at High-Water in any Place, ssS+ttL is a Maximum, and at Low-Water a Minimum or Ssds+Ltdt=o. But the inftantaneous Increment (ds) of the Sine of the Altitude of the Sun, is to the correfponding Increment (dz) of the Sun's diurnal Arc, as the Cofine (VI) of the Altitude of the Sun to Radius (1), or ds= VI-Xdz and the correfponding Decrement (dt) of the Sine of the Moon's Altitude, is to the corresponding Increment (dx) of the Moon's diurnal Arc, as the Cofine (VI-tt) of its Altitude to Radius (1), or -dt-dxXVI-tt=32dz XVI-tt, dx being to dz as 29 to 30, on account of the Motion of the Moon. Subftituting thofe Values of ds and dt in the Expreffion Ssds+Ltdt=o, we will have Ssy1-ss-32XL SV1-tt 29 L from whence it appears that at the Time

XtV1-it, or

=

tytt 30 S

The waters

every day.

of high and low Water the Quantities sy-ss and ty1-tt are always in the constant Ratio of 29 L to 30 S, or of 20 X5 to 30X2; but the Quantity 1-ss can never exceed; confequently tVI-it can

6

never exceed 30×1 or and of course one of the Factors t or v1-tt

29X 5

29 ;

must be always very small, which proves that the Moon is near the Meridian at High-Water, and near the Horizon at Low-Water.

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 Sun and Moon on the Sea, being greatest when the Moon is near the Meridian, it should 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 fhould be two Tides distant from each other, by the fame Interval that the Moon employs to pass 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 immediately follow the Appulfe of the Moon

to the Meridian.

XVI.

Since ty1-t 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 those 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 East to Weft every Day, thofe Luminaries move in a contrary Direction, the Sun 59 m. 8s.,3 the Moon 13 d. 10 m. 35s. in a Day, confequently after their Conjunction the Moon continually recedes 12 d. 11m. 268,7 from the Sun towards the Eaft each Day, until the 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 lets at Midnight. The following Days fhe comes tooner 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 he 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 Sur, rifes at Midnight, pafles at the Meridian at Six of the Clock in the Morning and fets at Noon. The following Day's the riles, palles at the Meridian, and fets fooner 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 rises above the Horizon, and by the diurnal Motion of the Earth which then confpires with that of the Moon. This Mo- What are tion impreffed on the Waters retains during fome Time its Acceleration, the Caufes fo that the Sea rifes higher and higher until the diurnal Motion of the which retard 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 usual 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.

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 manifeft that Low- does not water does not exactly fall between the two High-waters which immedi- exactly fall ately fucceed each other, but is fo much nearer to that which follows, as two Elevathe Elevation of the Pole in the proposed 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, after her Culmination. and why. 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 Re- ftrual Vavolution 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.

XVII.

XVIII.

riations.

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 at confpire to raise the Waters of the Sea. In the Quadratures on the full Moon.

the new and

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; tures. confequently the greatest Tides happen at full and new Moon, and the

The great

Tides do

not precifely happen

at that Time, and

why.

leaft at first and last Quarter.

XIX.

The greatest and leaft Tides do not happen in the Syfigies and Quaeft and leaft dratures, but are the Third or the Fourth in Order after the Syfigies and Quadratures, because as in other Cafes fo in this, the Effect is not the greatest or the least when the immediate Influence of the Cause is greatest or least. 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 greatest, they increase gradually, and the Waters have not acquired their greatest Height until fome Time after the Moon has passed the Sysigies, and the begins to counteract the Sun's Force and depress the Waters where the Sun raises them. Likewise the Tides which precede the Quadratures are not the leaft, they decreafe gradually and do not come to their leaft Height until some Time after the Moon has passed the Quadratures.

The great

on of the

Meridian

whilft the

dratures,

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 fooner ftracting from the external Caufes which retard it; by the combined after the Ap pulfe of the Forces of both must fall out in an intermediate Time, which corresMoon 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 pafles from Quadrature, this greatest Height answers more to the fetting of the the Syfigies Moon. The Sun in the first Case 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 passes answers more to the rifing of the Moon. In the first Case, 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 those 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

from the

to the Syfigies.

(b) See preceding Note