heres, (b) wherefore the annual Motion (P.) of the equinoctial Points of the Body compofed of the Ring and Globe to which it adheres, will be to the anaual Motion of the Nodes (N) of the Moon, in the compounded Ratio of 1436 X 4590 X 2 to 39343 X 424118.

But Newton found (Lem. 2. B. 3.), that if the Matter of the fuppofed Ring was fpread all over the Surface of the Sphere fo as to produce towards the Equator,the fame Elevation as that at the Equator of the Earth, the Force of the Matter thus fpread to move the Earth, would be lefs than the Force of the equatoral Ring in the Ratio of 2 to 5; therefore the annual Regrefs of the equinoctial Points is to the annual Regrefs of the Lunar Nodes, as 1436 X4590X2X2 to 39343×424118×5, and confequently in a Sydereal Year it will be 22, 58, 33 without any Regard being had to the Inclination of the Axis of the Ring, which Confideration caufes flill a Diminution in this Motion in the Ratio of the Cofine [91706] of this Inclination (which is 23 to the Radius (100000.)

The mean annual Preceffion of the Equinoxes produced by the Action of the Sun will be therefore 21' 6t nearly, fuppofing the Earth Homogeneous and the Depreffion towards the Poles

Simplon found from his Theory 21*6* (Miscellaneous Tracts) D'Alambert 23* nearly (Recherches Sur la Preceffion des Equinoxes) Euler 22 (Mem. de Berlin Tom. 5. 1749). And if this Quantity is greater by a third than what Obfervation indicates, it probably arifes from the Earth's not being Homogeneous, as was fuppofed, the Researches of Simpfon, Euler, and D'Alambert relative to this Object shall be explained hereafter.

IX.

Irregularities In this Manner Newton determined the mean Quantity of the Motion in the motiof the equinoctial Points. But not without examining the different Varie-on of the eties of the Action of the Sun on the protuberant Matter about the Equator points prosupposed to be reduced to a Ring.

quinoctial

duced by the

action of the

He fhews in Cor. 18, 19 and 20 of Prop. 66 that by the Action of the fun. Sun the Nodes of a Ring, fuppofed to encompas a Globe as the Earth, would reft in the Syfigies, in every other Place they would move in Antecedentia, they would move fwifteft in the Quadratures, that the Inclination of this Ring, would vary, that during each annual Revolution of the Earth, its Axe would Ofcillate, and at the end of each Revolution would return to its former Position, but that the Nodes would not return to their former Places, but would ftill continue to move in Antecedentia.

(b) Newton fupposes that the Sum of the Motions of the Ring and the Globe to which it adheres is to the Motion of the Ring, as the Revolution of the Nodes of this Ring is to the annual Motion of the Equinoctial Points of the Body composed of the Ring and Globe to which it adheres, in which he is mistaken as shall be fhewn hereafter.

The action.

of the fun

x.

The greatest Inclination of the Ring fhould happen when its Nodes are in the Syfigies, afterwards in the Paffage of the Nodes to the Quadratures, on the pro- this Inclination fhould diminish, and the Ring by its Effort to change its tuberant Inclination, impreffes a Motion on the Globe, and the Globe retains this Motion, till the Ring, or the protuberant Matter about the Equator, (for it is caufes an the fame Thing according to Newton) by a contrary Effort deftroys this annual nuta Motion, and impreffes a new Motion in a contrary Direction.

matter about

the equator

tion of the

axis of the earth.

Hence we fee that the Axis of the Earth fhould change its Inclination with Refpect to the Ecliptic, twice in its annual Course and return twice If the earth to its former Pofition. was elevated towards the

wards the

equator the

Newton has fhewn in Cor. 21 of Prop. 66 that the protuberant Matpoles and ter about the Equator making the Nodes retrograde, the Quantity of this depreffed to Matter increafing, this Regreffion, would increafe, and would diminish when this Matter diminished; hence if there was no Elevation towards the Equaequinoctial tor, there would be no Regreffion of the Nodes, and the Nodes of a Globe, which instead of been Elevated towards the Equator was depressed, and would ad- confequently would have its protuberant Matter about its Foles, would ftead of re- move in Confequentia.

points

vance in

trograding.

proves the

And he adds, (Cor. 22 of Prop. 66) that as the Form of the Globe Which enables us to judge of the Motion of the Nodes, fo from the Motion of dep effion of the Nodes we may infer the Form of the Globe; and confequently if the the earth Nodes move in Antecedentia, the Globe will be elevated towards the Equator, but on the Contrary depreffed, if the Nodes move in Confequentia, which is a further Proof of the Flatnefs of the Earth towards the Poles.

towards the

poles.

The moon contributes

to the pro

duction of

That the

action of the

XI.

We have hitherto confidered only the Action of the Sun in explaining the motion the Preceffion of the Equinoxes, and we have feen that in Confequence of of the equin this Action the equinoctial Points would receede annually 21 6. But oftial points. the Moon by her Attraction Acts on the Earth and influence very sensibly this Phenomenon, its Action being to that of the Sun as 2 to 1 (c) if the moon on the Inclination of its Orbit to the Equator was always the fame as that of Protuberant the Ecliptic to the Equator, the Regreffion thence refulting would be to that matter about arifing from the Sun's Action as 22 to 1. But because its Nodes shift conis more pow- tinually their Places,ithappens that the Inclination of its Orbit to the Equator, erful than on which depends its Effect varies continually, fo that when the afcending that of the Node is in Aries, the Inclination of the Moon's Orbit to the Equator a

the equator

fun.

(c) The Proportion of the Force of the Sun to that of the Moon, affigned by Newton is to 4, 4815. which he alfo affigns for the Proportion of the Preceffion of the Equinoxes produced by the Sun to that produced by the Moon but this Proportion does not agree with the Theories which depend on the Determination of the Mafs of the Moon, and it appears from Computation as fhall be fhewn hereafter, that the Preceffion of the Equinoxes produced by the Sun and that produced by the Moon are not in the fame Proportion as the Forces of thofe Lumminaries.

Nutation of

axis

mounts to 28d, but when the afcending Node nine Years after, is in Libra it scarce amounts to 18. in each Revolution, which renders the Preceffion arifing from the Action of the Moon very unequal during the Space of 18 Years, and Causes a Nutation in the Axis of the Earth, whereby its Inclination to the Ecliptic varies during the Revolution of the thesis of Nodes of the Moon; after which it returns to its former Pofition. This the earth Nutation from Theory, amounts to 19s, agreable to Obfervation, the produced by mean Preceffion arifing from the Action of the Moon, to 35, 5, confe- the action of quently the Preceffion arifing from the Action of the Sun to 14, 5, and the greatest Difference between the true Preceffion arifing from the Action of the Moon, and the mean Preceffion amounts to 173, 8.

Theory of the Ebbing and Flowing of the Sea.

the moon.

The expli

It is very easy to perceive the Connection between the Ebbing and Flow- cation of the ing of the Sea and the Preceffion of the Equinoxes. Newton deduces his Ex- ebbing and flowing of plication of the Ebbing and Flowing of the Sea, from the fame Corollaries of the fea, is Prop. 66, from whence we have seen he drew his Explication of the Precef- deduced fion of the Equinoxes; those two Phenomena are both one and the other a neceffary Confequence of the Attractions of the Sun and Moon on the Parts as is that which compose the Earth.

from prop.

66 & its cor.

of the pre ceffion of the

equinoxes. Error of

ebbing and

Galileo imagined that the Phenomena of the Tides might be accounted for, from the Motion of Rotation of the Earth, and its Motion of tranflation Galileo con round the Sun. But if this great Man had more attentively examined the cerning the Circumftances attending the Ebbing and Flowing of the Sea, he would have flowing of perceived that in Confequence of the diurnal Motion of the Earth, the Sea the fea. indeed would rife towards the Equator, and that the Earthwould affume the Form of a Spheroid depreffed towards the Poles, but this Motion of Rotation would never produce in the Waters of the Sea a Motion of Flux and Reflux, as Newton has demonftrated Cor. 19. Prop. 66. Newton Proves in this fame Corollary, applying what he had demonftrated in Cor. 5 and 6 of the Laws of Motion, that the Tranflation of the Earth round the Sun has no Effect on the Motion of Bodies at its Surface, and confequently the Motion of Translation of the Earth round the Sun, cannot Produce the Motion of Flux and Reflux of the Sea.

The ebbing On examining the Circumftances which attend the Ebbing and Flowing of and flowing the Sea, it was easy to perceive that those Phenomena depended on the Po- of the fea fition of the Earth with Refpect to the Sun and Moon; but it was not fo, to the action of discover the Manner those two Luminaries Produce those Phenomena and the fun and

arifes from

waters.

moon on the the Quantity that each contributes to their Production: we fee but the Effects in which the Actions of those two Luminaries are fo confounded, that it is only by the Affiftance of Newton's Principles we are enabled to distinguish one from the other, and affign their Quantity. It was referved for this great Man, to discover the true Caufe of the Ebbing and Flowing of the Sea, and to reduce those Causes to Computation; we fhall now trace the Road which conducted him to thofe Difcoveries.

Road which conducted

Newton to affign the quantity

that each of

IV.

He begins by examining in Prop. 66. the Principle Phenomena which fhould Refult from the Motion of three Bodies which attract each other mutually in the inverse Ratio of the Squares of the Distances, the fmall ones Revolving round the greater.

After having fhewn in the first 17. Corollaries of this Prop. the Irregularithofe lumi ties which the greater Body would Caufe in the Motion of the leffer, which naries contri itself revolves round the third, and by this Means having laid the Foundatiduce thofe on of the Theory of the Moon, he confiders in Cor. 18 feveral fluid Bodies phenomena. which revolve round a third, he afterwards fuppofes that thofe fluid Bodies

bute to pro

all become contiguous fo as to form a Ring revolving round the central Body, and proves that the Action of the greatest Body would produce in the Motions of this Ring the fame Irregularities as in those of the folitary Body in whofe Place the Ring was fubftituted; infine Cor. 19. he supposes the Body round which this Ring Revolves to be extended on every Side as far as this Ring, that this Body which is folid contains the Water of this Ring in a Channel cut all round its Circumference, and that it revolves uniformly round its Axis, he then proves that the Motion of the Water in this Channel will be accelerated and retarded alternately by the Action of the greater Body and that this Motion will be swifter in the Syfigies of this Water, and flower in its Quadratures, and finally that this Water will Ebb and Flow after the Manner of the Sea.

Newton applies this Prop. 66 and its Cor. to the Phenomena of the Sea (Prop. 24. B. 3.) and proves that they are a neceffary Confequence of the combined Actions of the Sun and Moon on the Parts which compofe the Earth.

v.

He afterwards investigates the Quantity, each of those Luminaries contribute, to the Production of thofe Phenomena. As this Quantity depends on their Distances from the Earth, the nearer they are to the Earth, the greater the Tides fhould be, Cæteris Paribus, when their Actions, confpire together: and according to Cor 14. Prop. 66, thofe Effects are in the Inverfe Ratio of the Cubes of their Distances from the Earth and the fimple Ratio of their Naffes.

Neu ton examines firft the Action of the Sun on the Waters of the Sea, because its Quantity of Matter with Refpect to that of the Earth is known. He obferves that the Attraction of the Sun on the Earth is counterbalanced

as to the Totality by the centrifugal Force arifing from the annual Motion of the Earth, which he confiders as uniform and circular: But what is true as to the Totality is not fo as to each particle of the Earth, that is,that the centri fugal Force of each of thofe Particles cannot be fuppofed equal to the Force with which the fame Particle is Attracted by the Sun, fince each Particle has the fame centrifugal Force, and the Particles of the Earth which are nearer the Sun are more attracted than those which are remoter. Thus the Dif tance of the Earth from the Sun, being 22000 Semidiameters of the Earth, and the Law of Attraction, the inverfe Ratio of the Squares of the Distances, the Attractive Force correfponding to the Point of the Earth neareft the Sun, to the Center of the Earth, and to the Point of the Earth remoteft from the Sun, will be nearly as 11001, 11000 and 10999, and as the Sun's Attraction balances the centrifugal Force of each Particle of the Earth, this Force will be Proportional to 11000; if from the attractive Force of the Sun on each of thofe three Points, the centrifugal Force be Subducted, there will remain 1,0,-1; which proves that the Center of the Earth is at Rest with Respect to the Motions of the Waters of the Sea, and that the two Extremities of the Diameter of the Earth directed towards the Sun, are actuated by equal Forces with oppofite Directions, whereby the Parts tend to recede from the Center of the Earth.

the fea.

If in the fame Diameter there be taken two Points equally distant from the First fourée Center, those two Points will be likewife actuated by equal Forces with op- of the ebpofite Directions, whereby they tend to recede from the Center; but this big and Force will decreafe as the Distance from the Center of the Earth. this Di- flowing of ameter of the Earth directed to the Center of the Sun may be called the Solar Axis of the Earth,if we now confider the Equator correfponding to this Axe,it is evident that each Point taken in the Plane of this Equator may be supposed equally diftant from the Center of the Sun, and confequently that none of the Points of this Plane are affected by the Inequality between the centrifugal Force and attractive Force,and confequently their Gravity towards the Center of the Earth will not be diminished, therefore if we conceive two Canals full of Water the one paffing thro' the demi folar Axe, and the other thro a Ray at its Equator, which communicate at the Center of the Earth, the Water will afcend in the first and descend in the other, this will happen both in the one and the other demi folar Axe, and is the firft Source of the Ebbing and Flowing of the Sea.

Each Particle of Water in the Canal of the demi folar Axe is attracted Second towards the Sun in the Direction of the Canal, but this Force acts on the fource of Particles of Water in the other Canal, obliquely, it therefore fhould be re- the ebbing and flowing folved into two, one perpendicular to the Canal, and the other parallel to it of the fea. The firft may be confidered as perfectly deftroied by the centrifugal Force; but the other Force adds to the Gravity of each Particle in this Canal, this