fun, produc

preffion of

the waters

fmall Force does not exist in the Canal of the demi folar Axe, and for this Reason the Water will defcend in the Canal of the folar Equator, and will fuftain that of the folar Axis to a greater Height. This is the fecond Source of the Ebbing and Flowing of the Sea.

From whence it appears that the Afcent of the Waters of the Sea does not arife from the total Action of the Sun, but from the Inequalities in that Action on the Parts of the Earth. Newton obferves that in Confequence of this Action the Figure of the Earth (abftracting from its diurnal Motion) ought to be an elliptic Spheroid having for greater and leffer Axes the folar Axe and the Diameter of its Equator, and determines in the following Manner the Force of the Sun which produces the difference of thofe Axes.

Determina He confiders the Figure of the Earth (abstracting from its diurnal Motion) tion of the rendered Elliptic by the Action of the Sun, as a fimilar Effect to the Figure force of the of the Orbit of the Moon, (abstracting from its excentricity) which he had ing theeleva fhewn (Prop. 66. Cor. 5) to be rendered Elliptic and to have its Center in the tion or de Center of the Earth, by the fame Action. He demonftrated (Prop. 25. B. 3) that the Force (F) which draws the Moon towards the Sun, is to of the fea in the centripetal Force (g) which draws the Moon towards the Earth, as the two points Square of the periodic Time (tt) of the Moon round the Earth, to the diametrially Square of the periodic Time (TT) of the Earth round the Sun, according to oppofite. Cor. 17 of Prop. 66; but the Inequality (V) in the Action of the Sun on the Parts of the Earth being to its Action (G), as the Ray (r) of the Earth, to the Ray (R) of its Orbit, and the Force (G) of the Sun which retains the Earth in its Orbit, being to the Force (g) which retains the Moon in its Or

R

bit, as TT Ray of the Earth's Orbit divided by the Square of its per

iodic Time, to

b

-

tt

-to

Ray of the Moon's Orbit divided by the Square of its periodic Time (Cor. 2 Prop. 4), VX G is to GX g, or the Inequality (V) in the Action of the Sun on the Parts of the Earth, is to the centripetal Force (g) of the Moon towards the Earth as rXR Rxb TT that is, as the Ray of the Earth divided by the Square of its periodic time round the Sun (TT) to the Ray of the Moon's Orbit, divided by the Square of its periodic Time round the Earth (→)

r

tt

Wherefore by the Compofition of Ratios, gXV is to FX g, or the Force (V) of the Sun disturbing the Motion of Bodies on the Surface of the Earth, is to its Force (F) with which it difturbs the Motion of the Moon, ΤΥΧΙ tt x b or as the Ray (r) of the Earth, to the Ray (b) of

as

TT

to

tt

the Moon's Orbit, that is, as I to 60

To compare now thofe two Forces with the Force of Gravity at the Surface of the Earth. Since the Force (F) which draws the Moon towards the Sun, is to the centripetal Force (g), which would retain the Moon in an Orbit, defcribed about the Earth quiefcent at its prefent Distance (60 Semidiameters of the Earth) as the Square of 27d. 7h. 43. to 365d. 6h. 9m, or as 1000 to 178725, or as 1 to 178 43; and that the Force which retains the Moon in its Orbit, is equal to the Force (7) which would retain it in an Orbit defcribed about the Earth quiefcent in the fame periodic Time, at the Distance of 60 Semidiameters, according to Prop. 60, in which it has been demonftrated that the actual Diftance (60 Semidiameters) of the Centres of the Moon and Earth, both revolving about the Sun, and at the fame Time about their common Centre of Gravity, is to the Distance (60 Semidiameters) of their Centres, if the Moon revolved about the Earth quiefcent in the fame periodic Time, as the Sum (1+42) of the Mafles of the Moon and Earth, to the firit of two mean Proportionals (42) between that Sum and the Mafs of the Earth. Confequently that the Force () which retains the Moon in its Orbit is lefs than the Force (g) which would retain it in an Orbit defccribed in the fame periodic Time, about the Earth quiefcent at the Distance 60 Semidiameters, in the Ratio of 60 to 60, (Cor. 2, P. 4); by the Compofition of Ratios FXg is to gXY or the Force (F) which draws the Moon towards the Sun, is to the centripetal Force (7) which retains the Moon in its Orbit, as 1X60 to 178 2x60. but this Force (7) which retains the Moon in its Orbit, (in approaching the Earth) increafing in the inverfe Ratio of the Square of the Diftance, is to the Force (G) of Gravity as 1 to 60x60, wherefore XF is to YXG, or the Force (F) which draws the Moon towords the Sun, is to the Force (G) of Gravity as 1X60 to 60X60×60×178 32 or as 1 to 638092,6.

of the action

ters of the fea to the force of gra

From whence Newton concludes [Prop. 36. B. 3.] that fince the Afcent of the Waters of the Sea, and the Elliptic Figure of the Lunar Orbit [ab- Proportion ftracting from its Excentricity] are fimilar Phenomena arifing from the Solar of the fun Force, and that in defcending towards the Surface of the Earth this Force on the wadecreases in the Ratio of 60 to 1. the Force of the Sun which depreffes the Waters of the Sea in the Quadratures, or at the Solar Equator, is to the Force of Gravity as I to 638092,6×60 or as I to 38604600. But this Force is double in the Syfiges, or in the Direction of the Solar Axis of what it is in the Quadratures, and acts in a contrary Direction [Cor. 6. Prop. 66], wherefore the Sum of the two Forces of the Sun on the Waters of the Sea, in the Quadratures and Syfigies, will be to the Force of Gravity as 3 to 38604600 or as I to 12868200. thofe two Forces united Compofe the total Force which raifes the Waters of the Sea in the Solar Canal, their Effect

vity.

Newton

concludes

theory that

the fun raif

es the water

of the fea

to a feet.

The ebbing and flowing of the fea

arifes from the motion

being the fame as if they were wholy employ'd in raising the Waters in the Syfigies, and had no Effect in the Quadratures.

VI.

Newton after having investigated the Force of the Sun which produces the Elevation of the Waters in the Solar Canal, determines in the following Manner the Quantity of this Elevation. He confiders the Elevation of the Waters of the Sea arifing from the Action of the Sun, as an Effect similar to the Elevation of the Equatorial Parts above the Polar Parts of the Earth, arifing from the centrifugal Force at the Equator. Now the centrifugal Force (C) at the Equator being to the Force of Gravity (G) at the Surface of the Earth as 1 to 289, and the Force of the Sun (F) exerted on the Waters of the Sea being to the Force of Gravity (G), as 1 to 12868200, by the Compofition of Ratios, FXG is to CXG, or the Force (F) of the Sun exerted on the Waters of the Sea, is to the centrifugal Force (C) at the Equator, as IX289 to 1X12868200 or as 1 to 44527; confequently the Flevation (85472 Feet) at the Equator produced by the centrifugal Force, is to the Elevation of the Waters in the Solar Canal produced by the Action of Sun, as I to 44527. which shews that the Elevation of the Waters in the Places directly under the Sun and in those which are directy oppofite to them is 1 Foot, 11, Inches.

VII.

The fluid Earth would preferve a Spheroidal form its longest Diameter pointing to the Sun without any Ebbing or Flowing of its Waters, if it had no Motion of Rotation. It is therefore the Rotation of the Earth round its of rotation Axis joined to its oblong Figure which caufes alternatly a Depreffion and of the earth Elevation of the Waters of the Sea. If the Axis of Rotation and the Solar Axis were the fame, the Waters of the Sea would have no Motion of reciprocation, because each Point during the Rotation of the Earth and moon. would be conftantly at the fame Distance from the Solar Poles.

and from

the actions of the fun

But

as those two Axes form an Angle, it is eafy to perceive that each Point of the Surface of the Earth approaches and recedes alternatly from the Solar Poles and that twice in a Revolution, and the Waters will continually rife in this Point during its Approach to, and will fall continually during its Recefs from thofe Poles. Newton inveftigated the Relation which fubfifts between the Method of Elevation of the Waters in any Place above that at the Solar Equator and eftimating the action of their Elevation in the Solar Canal; and found that the Square of the Radius the fun on [1] is to the Square of the Sine [ss] of the Altitude of the Sun in any of the fea in Place, as the Elevation [S] of the Waters in the Solar Canal to their Eleany place. vation [ss] in that Place.

the waters

VIII.

It is Manifeft that what has been faid with Refpect to the Sun fhould be applied without Restriction to the Moon and all the Phenomena of the Tides

the attrac

have fuch

prove evidently that the Action of this Luminary on the Waters is confidera- How is it bly greater than that of the Sun, which at first View should seem the more poffible that furprising, as the Attractive Force of the Sun arifing from its immenfe Bulk tion of the is fo powerful as to Force the Earth to Revolve round it, whilft the Irregu- moon can garities produced in its Orbit by the Action of the Moon are scarce fenfible, influence but if we confider that the Motion of the Sea proceedes from its Parts be- on the waing differently attracted from thofe of the reft of the Earth, becaufe their ters of the Fluidity makes them receive more easily the Impreffions of the Forces which fea & caufe Act on them, it will appear, that the Action of the Sun which is very pow- terations in erful on the whole Earth attracts all its Parts almost equally on Account of its the motion great Distance; but the Moon being much nearer the Earth Acts more une- of the earth. qually on the different Parts of our Globe, and that this Inequality should be much more fenfible than that of the Sun; thefe inequalities being in the Inverse Ratio of the Cubes of the Distances of the Luminaries from the Earth, and in the fimple Ratio of their Quantities of Matter.

in

The Elevation of the Waters of the Sea arifing from the Action of the Moon, in the Direction of the lunar Axis, above their Height at the lunar Equator, being once determined, the Elevation of the Waters of the Sea any Place above their Height at the lunar Equator, will be found, for in this Cafe, as in that of the Sun, the Square of the Radius (1) is to the Square of the Sine [tt] of the Altitude of the Moon in any Place, as the Elevation [L] of the Waters in the Direction of the lunar Axis, above their Height at the lunar Equator,to their Elevation [tt L] above the fame Height, in that Place.

IX.

fo little al

The varia tides arife

tions in the

fun and

From the Combination of the Actions of the Sun and Moon on the Waters of the Sea there refult two Tides, viz. the folar Tides and lunar Tides, which are produced independently of each other. Those two Tides by be- from the ing confounded with each other appear to Form but one, but fubject to great conjoint acVariations, for in the Syfigies the Waters are elevated and depreffed at the tions of the fame Time by both one and the other Luminary, and in the Quadratures the moon. Sun raises the Waters where the Moon depreffes them, and reciprocally the Sun depreffes the Waters where the Moon raises them, [one being in the Horifon when the other is at the Meridian] fo that from the Actions of those Luminaries fometimes confpiring and at other Times oppofed, there result very fenfible Variations both with refpect to the Height of the Tides and their Time.

X.

It is demonstrated that the Elevation of the Waters, produced by the conjoint Actions of the Sun and Moon, is fenfibly equal to the Sum of the Elevations produced by the Actions of each feperately, wherefore the whole Elevation produced by the united Actions of the two Luminaries will

ton came

of the moon

on the was ters of the

fea.

be Expreffed by ssS+ttL; which fhews that the Elevation of the Wa ters in any Place will continually increase until they attain their greatest Height, and then it is high Water, after which it will continually decreafe during fix Hours, and then it will be low Water; the Difference between those two Heights is called the Height of the Tide; from whence it appears that the Height of the Tides depends upon a great Number of Circumstances, viz. the Declination of each Luminary, the Age of the Moon, the Latitudes of Places and the Distance of the two Luminaries from the Centre of the Earth.

XI.

To examine the Variations in the Height of the Tides according to all thofe Circumstances, let us firft fuppofe the Orbit of the Moon and that of the Sun in the Plane of the Equator, and let us further fuppofe them How New- perfectly Circular, and let a Place be chofen at the Equator, in which Cafe we may fuppofe s=1 and 1, which will happen at the appulfe to estimate of the Luminaries to the Meridian in the Syfiges, and the whole Elevation the action will be expreffed by S+L; about fix Hours after so and to nearly and the Waters will have no Elevation confequently the Height of the Tides in the Syfigies will be expreffed by S+L; but in the Quadratures at the appulfe of the Moon to the Meridian t=1 and so, and the Elevation of the Waters will be expreffed by L, about fix Hours after s and to nearly, and the Elevation of the Waters will be evpreffed by S and the Height of the Tide will be expreffed by L-S, confequently the Height of the Tides in the Syfigies and Quadratures will be as S+L to L-S. if therefore the Height of the Tides in the Syfigies and Quadratures at the Time of the Equinoxes was determined from Obfervation, on the Coaft of an Ifland fituated near the Equator, in a deep Sea, and open on every Side to a great extent, the Ratio of L to S, the Effects of the Forces of the Sun and Moon, or the Ratio of thofe Forces which are proportional to thote Effects, would be found.

As no fuch Obfervations have been made, Newton employs for determining the Ratio of thofe Forces the Obfervations made by Sturmy three Miles below Briftol. this Author relates that the Height of the Alfcent of the Waters in the vernal and autumnal Conjunction and Oppofition of the Sun and Moon, amounts to about 45 Feet, but in the Quadratures to 25 only, wherefore L+S is to L-S as 45 to 25 or as 9 to 5, confequently 5L+5=9L-9S, or 14S-4L and S is to L as 2 to 7.

To reduce this Determination to the mean State of the variable Circumfances; it is to be obferved 1° that in the Syfigies the conjoint Forces of the Sun and Moon being the greateft,it has been fuppofed that the correfponding Tide is alfo the greateft, but the Force impreffed at that Time on the Sea being increased by a new Though a lefs Force fill acting on it until it be comes too weak to raife it any more, the Tides do not rife to their greatest Height but fome Time after the Moon has paffed the Syfigies, Newton