Abstracting from the

traction of

compounded Force and the fimple central Force which has no other Source but the Sun, is called the perturbating Force. The third Force is the deturbating Force, compounded of all the fame central Forces of the Planets reduced to one in a Direction perpendicular to the Planes of their Orbits; this Force is very small in comparison of the two others, on account of the fmall Inclination of those Planes to one another, and because the Sun placed in the Intersection of all those Planes does no way contribute to the Production of this deturbating Force. If the Planets were only actuated by mutual at the two firft Forces their Combination would serve to determine their the planets Trajectories which would be each in a conftant Plane, and if the perturtheir aphelia bating Force vanished then they would be regular Ellipfes, and confequentare at reft. ly the Aphelia and Nodes of the Planets would be fixed (Prop. 14. B. 3. & Prop. 1. & 11. B. 1.) if not; thefe Trajectories might be confidered as moveable Ellipfes on account of the prodigious excefs of the central Force of the Sun over the perturbating Force, it is thus Newton investigated the quantity and direction of the Motion of the Line of the Apfides of the Planets occafioned by the Action of Jupiter and Saturn, which according to his Determination follows the Sefquiplicate Proportion of the distances of the Planets from the Sun, from whence he concludes (Prop. 14. B. 3.) that fuppofing the Motion of the line of the Apfides of Mars in which this Motion is the most sensible to advance in a 100 Years 33m 20' in confequentia, The flow the Aphelia of the Earth, Venus and Mercury would advance 17m 40° motion of 10m 53 & 4m 16' respectively in the fame Time.

of the plan

the aphelia This flow Motion of the Aphelia confirms the Law of univerfal Graets is a new vitation, for Newton has demonftrated (Cor. 1. Prop. 45.) that if the proof that Proportion of the centripetal Force would recede from the Duplicate to apattraction proach to the Triplicate only the 60th Part, the Apfides would advance 3 inverfe ratio Degrees in a Revolution, therefore fince the Motion of the Apfides is al of the fquare most infenfible, Gravity follows the inverfe duplicate Proportion of the of the dif- diftances.

acts in the

tances

But the deturbating Force which acts at the fame Time caufes the Planes of those moveable Ellipfes to Change continually their Pofition; let there be supposed in the Heavens an immoveable Plane, in a mean Pofition between all thofe the Trajectory of the Earth would take in confequence of the deturbating Force, which may be called the true Plane of the Ecliptic, it is manifeft that this Plane being very little enclined to the Plane of the Orbit of Each Planet, it is almoft parallel to it, and consequently the Direction of the deturbating Force is always fenfibly perpendicular to the true Plane of the Ecliptic, and it is eafy to conceive that the effect of this Force produced in the Direction in which it acts, is either to remove the Planet from or to make it approach the true Plane of the Ecliptick, confequently to caufe a Variation in the Inclination of the fmall Arc which the Planet def

3

cribes that inftant with the true Plane of the Ecliptick, the Pofition of the Planes of the Trajectories of the Planets varies therefore in Proportion of the Intensity of the deturbating Force, and in the Direction in which this Force acts; if for Example the Force tends to make the Planet approach the true Plane of the Ecliptic the Node advances towards the Planet with a Velocity, which tho' fmall increases diminishes or vanishes according as the intensity of the deturbating Force increases diminishes or vanishes, but in this Cafe the Node cannot advance or go meet the Planet without moving in an oppofite Direction to that of the Planet, if therefore the heliocentric Motion is retrograde as in a great Number of Comets, that of the Nodes will be direct, the contrary would arrive if the deturbating Force tended to remove the Planet from the true Plane of the Ecliptic. Newton fays that fuppofing the Plane of the Ecliptic to be fixed the Regreffion of the Nodes is to the ton of the RetrogradaMotion of the Aphelium in any Orbit of a Planet as 10 to 21 nearly (c). nodes of the It is therefore only by this Compofition of Forces that all the Ir. planets according to regularities of the celeftial Motions can be inveftigated, it is by difcern- Newton ing the particular Effects of each of thofe compounded Forces, and afterwards uniting them, that not only thofe Irregularities that have been obferved can be determined, but thofe which will be remarked hereafter will be foretold. But it is easy to perceive how much fagacity and address to handle the sublimest Analysis these Reschearches require, and as it is almost impoffible to combine at once the central Forces of more than three Bodies placed in different Planes, in order to discover the irregularities of the Motions of a Planet or Comet it is necessary to calculate fuccefively the Variations that each Planet taken feperately can cause in the central Force of which the Sun is the Focus. The Succefs that has attended the united Efforts of the first Mathematicians in Europe shall be explained hereafter.

Theory of the Figure of the Planets.

I.

has not as

The Planets have another Motion viz. their Rotation round their Axes, The caufe we have seen already, that this Motion of Rotation has only been difcovered of the rotary in the Sun, the Earth, Mars, Jupiter and Venus, and that Aftronomers do motion of not agree about the Time in which Venus turns round tho' they are unani- the planets mous with respect to its Rotation. But tho' it has not been discovered from yet been dif Obfervation that Mercury, Saturn and the Satellities of Jupiter and Saturn covered. turn round their Axes, from the uniformity that Nature Obferves in her Operations, it is highly probable that those Planets revolve round their Axes, and that all the celestial Bodies partake of this Motion.

(e) De Systemate mundi Page 36 Edition, 1731.

The mutual attraction

pofe the

This Rotation of the Planets round their Axes is the only celeftial Motion which is uniform: this Motion does not appear to arise from Gravity, and its Caufe has not as yet been discovered.

II.

The mutual Attraction of the Parts of which the Planets are composed binds them together, and prevents their being difperfed by this Rotation, of the parts For it is well known that all Bodies moving round acquire a centrifugal Force which com by which they endeavour to recede from the Center of their Revolutions; hence, were not the Parts of the Planets held together by their mutual Atplanets prevents them tractions, they would be dispersed and scattered by their Rotation. For from being fuppofing the Gravity of any one Part of the furface of the revolving Body difperfed by destroyed, this Part instead of revolving with the Body would fly off in the direction of the tangent; therefore if Gravity did not counteract the Efforts of the centrifugal Force which the Parts of the celeftial Bodies acquire in revolving round their Axes, this force would disperse their Parts.

the rotation.

Tho' this mutual Attraction of the Parts of a Planet, counteracts the The rotato centrifugal Force, yet it does not destroy it, this Force ftill producing its ry motion, Effect, in rendering the diameters of the revolving Body unequal, fuppos eqators of ing it to be fluid; for the Planets being composed of Matter whofe Particles the planets. at equal Distances are equally urged to the Center, they would be exact

railes the

Newton pur

Spheres if they were at reft. But in confequence of the Motion of Rotation the Parts acquiring a centrifugal Force endeavour to recede from their Centers with Forces which increafe as they are placed nearer the Equator of the revolving Body, fince the centrifugal Forces of Bodies revolving in Circles, are as their Rays fuppofing the Time of Revolution to be equal; therefore fuppofing the Planets to be spherical and compofed of fluid Matter, before they acquired a Motion of Rotation, that the Equiblirium of their Parts may be preserved during this Rotation, and that they may affume a permanent form it was neceffary that the Column whose weight was dimin ifhed by the centrifugal Force fhould be longer than the Column whofe Weight is not altered by the centrifugal Force, and therefore the Equatorial Diameter must exceed the Diameter paffing thro' the Poles.

IV.

Newton in (Prop. 19. B 3) determines the excess of the equatorial Method above the polar Column of the Earth, fuppofing as he does all thro' the PrinJued for de- cipia that the Gravity of Bodies near the furface of the Earth is the result of te mining the Attraction, of all the Particles of which the Earth confidered as Homothe figure of geneous is composed: he employs for Data in the Solution of this Problem, the Earth. Ift the Semidiameter of the Earth confidered as a Sphere and determined by Picard to be 19615800. Feet 2, the Length of the Pendulum vibrating feconds in the Latitude of Paris which is 3 Feet 85 Lines.

From the Theory of Ofcillations and this Measure of a Pendulum vibrating feconds, he proves that a Body in the Latitude of Paris making the neceffary Correction for the refiftance of the Air, defçribes in a fecond 2174 Lines.

A Body revolving in a Circle at the Distance of 19615800 Feet from the Center, which is the Semidiameter of the Earth, in 23h 56" 4' which is the exact Time of the diurnal Revolution, fuppofing its Motion uniform, defcrbes in a fecond, an Arc of 1433, 46 Feet; of which the verse, Sine is, 0,0523656 Feet, or 7, 54064 Lines; therefore in the Latitude of Paris the Force of Gravity is to the centrifugal Force, which Bodies at the Equator derive from the diurnal Rotation, as 2174 to 7, 54064. Adding therefore to the Force of Gravity, in the Latitude of Paris, the Force detracted therefrom by the centrifugal Force in that Latitude, in order to obtain the total Force of Gravity in the Latitude of Paris, Newton finds that this total Force is to the centrifugal Force under the Equator as 289 to 1 fo that under the Equator the centrifugal Force diminishes the centrifugal Force by

Newton determines (Cor. 2. Prop. 91.) the Proportion of the Attraction of a Spheroid upon a Corpufcule placed in its Axe produced, to that of a Sphere, on the fame Corpufcule, whofe Diameter is equal to the leffer Axe of the Spheroid; employing therefore this Proportion and fuppofing the Earth homogeneous and at reft, he finds (Prop. 19. B. 3.) that if its Form be that of a Spheroid whofe leffer Axe is to the greater as 100 to 101, the Gravity (g) at the Pole of this Spheroid will be to the Gravity (Y) at the Pole of a Sphere, whofe Diameter is the leffer Axe of the Spheroid as 126 to 125.

In the fame Manner fuppofing a Spheroid whofe equatorial Diameter is the Axe of Revolution, the Gravity (V) at the Equator which is the Pole of this new Spheroid, will be to the Gravity (F) of a Sphere at the fame Place having the fame Axe of Revolution, as 125 to 126.

Newton fhews afterwards that a mean proportional (G) between these two Gravities (V, I) expreffes the Gravity at the Equator of the Earth: confequently the Gravity (G) at the Equator of the Earth, is to the Gravity (I) of a Sphere at the fame Place, having the fame Axe of Revolution, as 125, to 126. and having demonftrated (Prop. 72) that the Attraction of homogeneous Spheres at their Surfaces is proportional to their Rays, it follows that the Gravity (2) at the Surface of the Sphere whofe Diameter is the leffer Axe of the Spheriod, is to the Gravity () at the Surface of the Sphere whole Diameter is the great Axe of the Spheroid, as 100 to 101 wherefore by the Compofition of Ratios g X Y X T is to YX GX or the Gravity (g) of the Earth, at the Pole, is to the Gravity (G) at the Equator as 126 X 126 X 100 to 125 × 125 × 101 that is as 501 to 500.

But he had demonstrated, (Cor. Prop. 91.) that if the Corpufcule is placed within the Spheroid, it would be attracted in the Ratio of its diftance

From

whence he

from the Center; therefore the Gravities in each of the Canals correfponding to the Equator and to the Pole will be as the Distances from the Cen ter of the Bodies, which are placed in thofe Canals; therefore fuppofing thefe Canals to be divided into Parts, proportional to the Wholes, confequentely at Distances from the Center proportional to each other, by Tranfverfe Planes, which pafs at Distances proportional to thofe Canals. The Weights of each Part in one of those Canals, will be to the Weights of each correfpondent Part in the other Canal, in a constant Ratio, confequently these Weights will be to each other in a conftant Ratio of each Part, and their accelerative Gravities Conjointly, that is as 10 to 100, and 500 to 501, that is, as 505 to 501; therefore if the centrifugal Force of any Part of the Equatorial Canal be to the abfolute Weight of the fame Part as 4 to 505, that is, if the centrifugal Force detracts from the Weight of any Part of the Equatorial Canal Parts, the Weights of the Correfpondent Parts of each Canal will become equal, and the Fluid will be in Equilibrio. But we have seen that the Centrifugal Force of any Part under the Equator, is to its Weight as I to 289, and not as 4 to 505; the Proportion of the Axes therefore must be different from that of 100 to 101, and fuch a Proportion must be found as will give the Centrifugal Force under the Equator, only the 289th Part of Gravity.

But this is eafly found by the Rule of Three; for if the Proportion concludes of 100 to 101 in the Axes has given that of 4 to 505 for the Prothe ratio of portion of the Centrifugal Force to Gravity, it is manifest that the Prothe axes of portion of 229 to 230 is requifite to give the Proportion to 289 of be that of the Centrifugal Force to Gravity.

the earth to

229 to 230.

The flat

wards the

But

This Conclufion of Newton, that is, the Quantity of the Depreffion of nefs or the the Earth towards the Poles, which he has determin'd is grounded on earth to- his Principle of the mutual Attractions of the Parts of Matter. poles would this Depreffion towards the Poles would alfo refult from the Theory. always re- of Fluids, and that of Centrifugal Forces, tho' Newton's Difcoveries fult from the concerning Gravity werè rejected, unlefs very improbable Hypothefes contheory of centrifugal cerning the Nature of primitive Gravity were adopted.

forces and

that of fluids what hypo

VI.

Notwithstanding the Authority of Newton, and although Hugbens in thefis of gra affuming a different Hypothefis of Gravity arrived, at the fame Conclufion vity is af- of the Depreffion of the Earth towards the Poles; and tho' all the Exfunied. periments made on Pendulums in the different Regions of the Earth, The mea- confirmed the decrease of Gravity towards the Equator, and confefure of the quently favoured the opinion of the Flatnefs of the Earth towards the the meridi Poles, yet the Meafures of Degrees in France, which feemed to dean in France creafe as the Latitude increafed ftill rendered the Figure of the Earth.

degrees of