An Inquiry concerning the Figure of Such Planets as Revolve about an Axis, Supposing the Density Continually to Vary, from the Centre towards the Surface; By Mr. Alexis Clairaut, F. R. S. and Member of the Royal Academy of Sciences at Paris. Translated from the French by the Rev. John Colson Lucas. Prof. Math. Cantab. and F. R. S.

I. An Inquiry concerning the Figure of such Planets as revolve about an Axis, supposing the Density continually to vary, from the Centre towards the Surface; by Mr. Alexis Clairaut, F. R. S. and Member of the Royal Academy of Sciences at Paris. Translated from the French by the Rev. John Colson Lucas. Prof. Math. Cantab. and F. R. S. Notwithstanding that Part of Sir Isaac Newton's Mathematical Principles of Natural Philosophy, where he treats of the Figure of the Earth, is deliver'd with the usual Skill and Accuracy of that great Author; yet I thought something farther might be done in this Matter, and that new Inquiries may be proposed, which are of no small Importance, and which possibly he overlook'd, through the Abundance of those fine Discoveries he was in Pursuit of. What at first seem'd to me worth examining, when I apply'd myself to this Subject, was to know why Sir Isaac assumed the Conical Ellipsis for the Figure of the Earth, when he was to determine its Axis? For he does not acquaint us why he did it, neither can we perceive how he had satisfied himself in this Particular: And unless we know this, I think we cannot intirely acquiesce in his Determinations of the Axes of the Planets. It seems as if he might have took any other oval Curve, as well as the conical Ellipsis of Apollonius, and then he would have come to other Conclusions about those Axes. I began then with convincing myself by Calculation, that the Meridian of the Earth, and of the other Planets, is a Curve very nearly approaching to an Ellipsis; so that no sensible Error could ensue by supposing it really such. I had the Honour of communicating my Demonstration of this to the Royal Society, at the Beginning of the last Year; and I have since been inform'd, that Mr. Stirling, one of the greatest Geometers I know in Europe, had inserted a Discourse in the Philosophical Transactions, No. 438, wherein he had found the same thing before me, but without giving his Demonstration. When I sent that Paper to London, I was in Lapland, within the frigid Zone, where I could have no Recourse to Mr. Stirling's Discourse, so that I could not take any Notice of it. The Elliptical Form of the Meridian being once proved; I no longer found any thing in Sir Isaac Newton, about the Figure of the Earth, which could create any new Difficulty; and I should have thought this Question sufficiently discus'd, if the Observations made under the Arctic Circle had not prevail'd on us to believe, that the Shape of the Earth was still flatter than that of Sir Isaac's Spheroid; and if he himself had not pointed at the Causes, which might make Jupiter not quite so flat, as by his Theory, and the Earth something more. As to Jupiter, he says, (Page 416 of the 3d Edition of Phil. Nat. Prin. Math.) that its Equator consists of denser Parts than the rest of its Body, because its Moisture is more dried up by the Heat of the Sun. But as to the Earth, he suspects its Flatness to be a small matter greater than what arises by his Calculation. lation. He insinuates, that it may possibly be more dense towards the Centre than at the Superficies. (See Art. XXIV. following) I am something surprized that Sir Isaac should imagine, that the Sun's Heat can be so great at Jupiter's Equator, when it has no such Effect at that of the Earth; and that he does not ascribe each to a like Cause, by supposing also, that Jupiter may be of a different Density at the Centre from that at the Superficies. But whatever Reason he might have for introducing two different Causes, I give the Preference to the Hypothesis which supposes unequal Densities at the Centre and at the Circumference. I have inquired, by the Assistance of this Theory, what would be the Figure of the Earth, and of the other Planets which revolve about an Axe, on Supposition that they are composed of similar Strata, or Layers, at the Surface; but that their variable Density, from the Centre towards the Circumference, may be expounded by any Algebraical Equation whatsoever. And though my Hypothesis should not be conformable to the Laws of Nature, or even though it should be of no real Use (which would be the Case, if the Observations made by the Mathematicians now in Peru, compared with ours in the North, should require that Proportion of the Axes, which is derived from Sir Isaac's Spheroid); I thought however that Geometricians would be pleased with the Speculations contain'd in this Paper, as being, if not useful, yet curious Problems at least. PART I. In which are found the Laws of Attraction, which are exerted upon Bodies at a Distance, by a Spheroid compos'd of Orbs of different Degrees of Density. PROBLEM I. To find the Attraction which a homogeneous Spheroid BNEbE, (Tab. 1. Fig. 1.) differing but very little from a Sphere, exerts upon a Corpuscle placed at A in the Axis of Revolution. I. We may conceive the Space BNEbDMB, included between the Spheroid and the Sphere, to be divided into an infinite Number of Sections perpendicular to the Axe ACb. Supposing then that every one of the Particles, which are contain'd in one of these Elements or Moments NnmM, exerts the same Quantity of Attraction upon the Body at A, which may be suppos'd because of the Smallness of NM; we shall have $c \alpha \times PM^2 \times Pp \times \frac{AP}{AM^3}$ for the Attraction of any one of these Elements; putting c for the Ratio of the Circumference to the Radius, and $\alpha$ for the given Ratio of MN to PM, that is, of DE to CD. Now if we make CA = e, CB = r, AM = z; and for PM, AP, Pp, if we substitute their Values express'd by z, and then seek the Fluent of the foregoing Quantity; we shall have $\frac{4car^3}{3ee} - \frac{4car^5}{5e^4}$ for the Value Value of the whole Attraction of the Solid generated by the Revolution of BDbEB: To which if we add $\frac{2r^3c}{3ee}$, the Attraction of the Sphere, we shall have $\frac{2r^3c}{3ee} + \frac{4cr^3a}{3ee} - \frac{4car^5}{5e^4}$ for the required Attraction of the Spheroid upon the Corpuscle A. **Problem II.** Supposing now the Spheroid Bebe (Fig. 2.) to be no longer of a homogeneous Matter, but to be composed of an infinite Number of Elliptical Strata, all similar to BEb, the Densities of which are represented by the Ordinates KT of any Curve whatever VT, of which we have the Equation between CK and KT; the Attraction is required which this Spheroid exerts upon a Corpuscle placed at the Pole B. II. Making BC = e, CK = r, by the foregoing Proposition, we should have $\frac{2r^3c}{3ee} + \frac{4cr^3a}{3ee} - \frac{4car^5}{5e^4}$ for the Attraction of the Spheroid KLK, if it consisted of homogeneous Matter; and the Fluxion of this Quantity $\frac{2rrcr}{ee} + \frac{4car^2r}{ee} - \frac{4car^4r}{e^4}$ would be the Element or Moment of the Orb KLKklk. But because the Density is variable, we must multiply this Value of the Attraction of the Orb by KT, and the Fluent of this Quantity will be the Value of the Attraction of the Spheroid KLK. As to the Value of KT, which expresses the Density of the Stratum or Bed KLKklk, we shall take only only $fr^p + gr^q$, because we shall see afterwards, that a Value more compounded, at $fr^p + gr^q + hr^s + ir^t$, &c. which by the Property of Series may express all Curves, would not produce any Variety in the Calculation. Therefore multiplying the foregoing Equation by $fr^p + gr^q$, we shall have $$\frac{2cf \times r^{3+p}}{eex^{3+p}} - \frac{4cafr^{5+p}}{e^4 \times 5+p}$$ $$+ \frac{2cg \times r^{3+q}}{eex^{3+q}} - \frac{4cagr^{5+q}}{e^4 \times 5+q}$$ for the Quantity of Attraction of the Spheroid KLK, exerted upon a Corpuscle placed at B. III. In this Value making $r = e$, we shall have $$\frac{2cfe^{1+p}}{3+p} + \frac{8cfe^{1+p_a}}{3+p \times 5+p} + \frac{2cge^{1+q}}{3+q} + \frac{8cge^{1+q_a}}{3+q \times 5+q},$$ which will express the Force of Attraction of the Spheroid BEb, exerted upon a Corpuscle placed at the Pole B. **Theorem.** A Corpuscle being placed in any Point N of the Surface of the foregoing Spheroid BEbe, I say it will undergo the same Attraction from this Spheroid, as if it were placed at the Pole N of a second Spheroid revolving about the Axe NO, the second Axe being the Radius of a Circle equal in Superficies to the Ellipsis FG; supposing this second Spheroid NGOF (Fig. 3.) to be composed of the Strata Mm qQ, whose Densities are the same as those of the Strata Kk Ll Kk, of the first Spheroid. IV. In IV. In the Discourse I had the Honour of communicating to the Royal Society, being then at Torneo, printed in the Philosophical Transactions; No 445. I have demonstrated this Proposition as to a homogeneous Spheroid; and the same Reasoning will obtain in this Case also. Problem III. To find the Attraction which the Spheroid Bebe (Fig. 2.) exerts upon a Corpuscle placed at any point N of the Superficies. V. We will make, as above, BC = e, CE = e + eα, and also CN = e + eλ, and half the Conjugate Diameter of CN will be CG = e + eα - eλ; whence the Radius of a Circle, equal in Superficies to the Ellipsis E.G., will be a mean proportional between CE and CG, that is to say, e + eα - \(\frac{1}{2}e\lambda\). Therefore the Spheroid Bebe exerts the same Attraction at N, as would be exerted at the Pole of a Spheroid NGO F, (Fig. 3.) of which the principal Axis would be NO = 2e + 2eλ, and the second would be to the Principal as \(1 + \alpha - \frac{3}{2}\lambda\) to 1. Therefore in the Expression of the Attraction at the Pole, (Art. III.) we must substitute \(e + e\lambda\) instead of e, and \(\alpha - \frac{3}{2}\lambda\) instead of α. But if f and g must no longer be the same; for we may easily perceive by the foregoing Theorem, that the Density must be the same in this Spheroid NGO F, at the Distance r + rλ from the Centre, as it is in the Spheroid Bebe at the Distance Distance r. Therefore \( f \left( \frac{e}{1 + \lambda} \right)^p + g \left( \frac{e}{1 + \lambda} \right)^q \) must be put instead of \( fe^p + ge^q \). Thus we shall have \[ \frac{2cfe^{1+p}}{3+p} + \frac{2p-2cfae^{1+p}}{3+p \times 5+p} + \frac{8cfae^{1+p}}{3+p \times 5+p} + \frac{2cge^{1+q}}{3+q} \] \[ + \frac{2q-2cgaec^{1+q}}{3+q \times 5+q} + \frac{8cgaec^{1+q}}{3+q \times 5+q} \] for the Attraction of the Spheroid BE be at N. VI. If we make \( \lambda = \alpha \), the foregoing Expression will be reduced to this \[ \frac{2cfe^{1+p}}{3+p} + \frac{2cfe^{1+p}}{5+p} + \frac{2cge^{1+q}}{3+q} + \frac{2cge^{1+q}}{5+q}, \] which expresses the Attraction of the Equator. VII. If we would have the Attraction at any Point M within the Spheroid, in the Expression of the Attraction at N, we must put r instead of e. The Proof of this is plain from the same Reasons that Sir Isaac Newton makes use of, (Corol. 3. Prop. XCI. L. I. Princip. Math.) to shew that the Attraction of an Elliptic Orb, at a Point within it, is none at all. **Problem IV.** Let \( R \Pi r \pi \) (Fig. 4.) be a Circle whose Centre is Y; 'tis required to find the Attraction which this Circle exerts upon a Corpuscle at N, according to the Direction HY; supposing the Point H, which answers perpendicularly below the Point N, to be at a very small Distance from the Point Y. VIII. Let there be drawn \( \Pi H \pi \) perpendicular to the Diameter \( R Y r \), and let the Space \( R \Pi \pi \) be transferr'd fer'd to \( \pi \Pi Z \). Then the Space \( \pi Z \Pi r \) will be the only Part of \( R \Pi r \pi \), which will attract the Body N according to HY. To find the Attraction of this little Space, we will suppose it to be divided into the Elements TtsS, the Attractions of which, according to HY, will be \( \frac{TtsS \times QT}{NT^3} \), or \( \frac{2HY \times Qq \times QT}{NT^3} \), the Fluent of which \( \frac{2HY \times HQ \times TZ}{NT^3} \) is the Attraction of TZrS, according to HY. In which if we put \( \Pi \pi \) for HQ, we shall have \( \frac{\Pi H \pi R \times 2HY}{NT^3} \), or \( \frac{1}{2} \frac{HY \times \Pi H^2 \times c}{NT^3} \), for the Attraction required. IX. It is easy to perceive, that if, instead of a Circle, the Curve \( R \Pi r \) were an Ellipsis, or any other Curve whose Axes were but very little different from one another, the foregoing Solution would be still the same. **Problem V.** *To find the Attraction which an Elliptical Spheroid KLk (Fig. 5.) exerts upon a Corpuscle placed without its Surface at N, according to the Direction CX perpendicular to CN.* X. To perform this, we will begin by drawing the Diameter \( C \mu v \), which bisects the Lines \( R r \) perpendicular to \( CN \); and the Ratio of \( CH \) to \( HY \) shall be call'd n. Then esteeming the Ellipsis \( R r \) as a Circle, (see the foregoing Article) we shall have by the Problem aforegoing \( \frac{1}{2} nc \times RH^2 \times CH \) for its Attraction. according to HY; which being multiply'd by the Fluxion of MH, the Fluent of this will be the Attraction of the Segment of the Spheroid R Mr. This Calculation being made, and Nm being substituted for NR, we shall have $\frac{2cnr^5}{5e^4}$ for the Attraction of the Spheroid in N, according to the Direction CX. **Problem VI.** *To find the Attraction of a Corpuscle N, according to CX, towards an Ellipsoid BNEbe, composed of Strata, the Densities of which are defined by the Equation $D = fr^p + gr^q$.* XI. Take the Fluxion of the Quantity $\frac{2cnr^5}{5e^4}$, which expresses the Attraction of the homogeneous Ellipsoid KLk, and you will have $\frac{2cnr^4r}{e^4}$ for the Attraction of an infinitely little elliptick Orb; which being multiply'd by the Density D, gives $\frac{2cnfr^4+p}{e^4} + \frac{2cgnr^4+qr}{e^4}$, the Fluent of which $\frac{2cfnr^5+p}{5+pxe^4} + \frac{2cgnr^5+q}{5+qxe^4}$, is the Attraction of the Spheroid KLk, according to CX. Therefore the total Attraction of the Spheroid BNEbe upon the Corpuscle N, according to the Direction CX, will be $\frac{2cfne^1+p}{5+p} + \frac{2cgne^1+q}{5+q}$. Now Now if we have regard to the Smallness of the Line \( N \nu \), and observe how little the Angle \( \nu NC \) will differ from a right one, we may perceive that the Diameter \( CN \) contains the same Angle with the perpendicular \( NX \) in \( N \), as the Diameter \( CN \) with the perpendicular at \( \nu \); that is to say, that the Angle \( NC \nu \) is the same as the Angle \( CNX \); so that instead of \( n \) we may take \( \frac{CX}{CN} \). Wherefore the foregoing Expression of the Attraction of the Ellipsoid \( BEbe \), acting according to the Direction \( CX \) upon a Corpuscle placed in \( N \), will be \( \frac{2cfe^{1+p}}{5+p} \times \frac{CX}{CN} + \frac{2cge^{1+q}}{5+q} \times \frac{CX}{CN} \). **Problem VII.** *To find the Direction of the Attraction of a Corpuscle \( N \) towards the Ellipsoid.* XII. By the second Problem we shall find the Attraction of the Spheroid according to \( CN \) to be \( \frac{2cfe^{1+p}}{3+p} + \frac{2cge^{1+q}}{3+q} \), by expunging what may be here expunged. Then by taking a fourth proportional to these three Quantities, the first of which is the Attraction according to \( CN \), the second is that according to \( CX \), and the third is the right line \( CN \); there will arise \[ \frac{fe^{1+p}}{5+p} + \frac{ge^{1+q}}{5+q} \times CX = CL. \] \[ \frac{fe^{1+p}}{3+p} + \frac{ge^{1+q}}{3+q} \] O o z Whence Whence we shall have NI for the Direction requir'd, of the Attraction of the Corpuscle N. XIII. If we suppose \( p = q = 0 \), that is, if the Spheroid be homogeneous, we shall have \( CI = \frac{3}{5} CX \); which agrees with what Mr. Stirling has found, in that curious Dissertation he has publish'd in the Philosophical Transactions, No 438. PART II. The Use of the foregoing Problems, in finding the Figure of Spheroids, which revolve about an Axis. XIV. Let us now suppose, that the foregoing Spheroid BNE bc, (Fig. 5.) which is still composed of Beds or Strata of different Densities, revolves about its Axis Bb, and that it is now arrived at its permanent State. It is plain that the Particles of the Fluid, which are upon its Surface, must gravitate according to a Direction perpendicular to the Curvature BNE; for without this Condition there could be no Equilibrium. We shall now inquire, whether the Elliptic Figure we have ascribed to our Spheroids can have this Property, and to produce this Effect, what must be the Relation between the Time of Revolution of the Spheroid and the Difference of its Axes. Let us then put \( \Phi \) for the centrifugal Force at the Equator, and the centrifugal Force at N will be \( \frac{\phi \times PN}{CE} \), or \( \frac{\phi \times Cx}{2CE \times a^2} \), because \( 2PN \times a = Cx \). By resolving this centrifugal Force according to the Perpendicular to CN, we shall have $\frac{\phi \times CX}{2a \times CE}$; which being added to $\frac{2cfe^1 + p}{5 + p} \times \frac{CX}{CN} + \frac{2cge^1 + q}{5 + q} \times \frac{CX}{CN}$ found by Prob. V. will give the whole Force of the Body N, according to the Direction CX, when the Spheroid is converted about its Axis. But because this Body, by virtue of the Attraction according to CN, and the Force according to CX, ought to have a perpendicular Tendency to the Superficies; we shall have this Analogy, CN. CX : $\frac{2cfe^1 + p}{3 + p} + \frac{2cge^1 + q}{3 + q} \times \frac{CX}{CE} + \frac{2cfe^1 + p}{5 + p} \times \frac{CX}{CN} + \frac{2cge^1 + q}{5 + q} \times \frac{CX}{CN}$. And hence, because CN and CE may be assumed as the same on this Occasion, it will be $$\phi = \frac{8cfe^1 + pa}{3 + p \times 5 + p} + \frac{8cge^1 + qa}{3 + q \times 5 + q}.$$ And as in this Value of the centrifugal Force, no Quantity enters but what will agree to any Point N; we may therefore conclude, that when our suppos’d elliptical Spheroid performs its Rotation in a proper Time, so that the centrifugal Force at the Equator may be as before; then the centrifugal Force in any other Place N will be such as it ought to be, to cause Bodies to gravitate in a perpendicular Direction to the Surface. The Spheroid being suppos’d elliptical, Bodies will gravitate perpendicularly to its Surface. XV. If XV. If we now consider, that ED (Fig. 6.) being taken for the centrifugal Force in E, then will MN express the centrifugal Force in N, and consequently MI will be such a Part of this Force as acts according to NC; we shall have \(\frac{8cfe^1 + p}{3 + p \times 5 + p} + \frac{8cge^1 + q}{3 + q \times 5 + q}\) to be subtracted from the Attraction at N. Hence \(\frac{2cfe^1 + p}{3 + p} + \frac{2p - 10cf \lambda e^1 + p}{3 + p \times 5 + p} + \frac{8cfae^1 + p}{3 + p \times 5 + p} + \frac{2cge^1 + q}{3 + q} + \frac{2q - 10cgae^1 + q}{3 + q \times 5 + q} + \frac{sczae^1 + q}{8 + q \times 5 + q}\) will be the Gravity at N. XVI. In this Value making \(\lambda = \alpha\), we shall have \(\frac{2cfe^1 + p}{3 + p} + \frac{2p - 2cfae^1 + p}{3 + p \times 5 + p} + \frac{2cge^1 + q}{3 + q} + \frac{2q - 2cgae^1 + q}{3 + q \times 5 + q}\) for the Gravity at the Equator. XVII. If we subtract the Value of the Gravity in N from the Value of the Attraction or Gravity at the Pole, (Art. III.) we shall have \(\frac{10 - 2pcf \lambda e^1 + p}{3 + p \times 5 + p} + \frac{10 - 2qcg \lambda e^1 + q}{3 + q \times 5 + q}\). But it is easy to perceive, that \(\lambda\) is proportional to the Square of the Sine of the Arc PM, or of the Complement of the Latitude. Whence we may therefore conclude, that the Diminution nution of the Gravity from the Pole to the Equator is proportional to the Square of the Cosine of the Latitude; or, which is the same thing, that the Augmentation of Gravity from the Equator to the Pole is as the Square of the Sine of the Latitude, as Sir Isaac Newton has demonstrated in his Hypothesis of a homogeneous Spheroid. XVIII. From the following Calculation it is easy to conclude, that Sir Isaac's Theorem, (Prin. Math. L. III. Prop. XX.) which is this, that the Gravity in any Place within is reciprocally as the Distance from the Centre, cannot obtain here. For we may see by the foregoing Expression, that the Gravity in N cannot be to the Gravity in P as 1 to $1 + \lambda$, except when $p = q = 0$, which happens only in Sir Isaac's homogeneous Spheroid. It was for want of considering, that this Theorem was demonstrated by Sir Isaac only in the Case of his homogeneous Spheroid, that several Geometricians have too hastily concluded; this Theorem might be apply'd to determine the Ratio of the Earth's Axes, and the Lengths of the Pendulum observed in two Places of different Latitudes. Dr. Gregory is one of those who have fallen into this Mistake, in his Elements of Astronomy, Lib. III. Sect. 8. Prop. 52. And in the Philosophical Transactions, No 432. it is concluded, from the Proportion of Gravity at Jamaica to that at London, that the Diameter of the Equator must exceed the Earth's Axis by $\frac{1}{190}$th Part, which Computation was founded on this 20th Proposition, Lib. III. of Sir Isaac's Principia, which is true only of his Spheroid. XIX. Let XIX. Let us now suppose, that the centrifugal Force at the Equator is known by Observation, as also within the Earth, &c. and that it is a certain Part $\frac{1}{m}$ of the Gravity; by Articles XIV. and XVI. we shall have this Equation: $$\frac{2cfe^1 + p}{3 + p} + \frac{2v - 2cfe^1 + p_a}{3 + p \times 5 + p} + \frac{2cge^1 + q}{3 + q} + \frac{2q - 2cge^1 + q_a}{3 + q \times 5 + q} = \frac{8cfme^1 + q_a}{3 + p \times 5 + p} + \frac{8cmge^1 + q_a}{3 + q \times 5 + q}.$$ From hence it will be easy to derive the Value of $\alpha$, because $f, g, p, q$, will be given, from the Hypothesis that will be chosen, for the Variation of the Density in the internal Parts of the Spheroid. XX. And if on the contrary $\alpha$ be given, that is, if we know by Observation the Ratio of the Axes of the Planet concern'd; then by the foregoing Equation we may perceive, whether we have assumed an agreeable Hypothesis for the Variation of the Densities: But we cannot precisely determine what this Hypothesis must be, because there is but one Equation, in which four indeterminate Quantities $f, g, p, q$, are involved. And indeed there might be many more than four indeterminate Quantities, if we should assume more than two Terms in the general Equation of the Densities $D = fr^p + gr^q + hr^s$, &c. XXI. In order to apply the foregoing Theory to the Earth, it might seem at first Sight, that by the Assistance of Observations made for measuring the Length of of the Pendulum, we might have other Equations, which with the foregoing Equation A, would determine the Coefficients & Exponents now mention'd; but we shall soon see the Impossibility of this upon two Accounts: First, There need be only two Observations, as to what concerns the Length of the Pendulum. For because by Art. XVII. the Augmentation of the Gravity from the Equator to the Pole is proportional to the Square of the Sine of the Latitude, two Observations as much determine the Problem as an infinite Number can do: So that we could have but one other Equation besides the foregoing. This Equation will \[ \frac{5-pf\alpha}{3+p\times5+p} + \frac{5-qg\alpha}{3+q\times5+q} \] be (B) \(p-p = \frac{5-pf\alpha}{3+p\times5+p} + \frac{f}{3+p} + \frac{g}{3+q} + \frac{q-1g\alpha}{3+q\times5+q}\) The first Member of this Equation expresses the Gravity at the Equator subtracted from the Gravity at the Pole, and divided by the Gravity at the Equator; a Quantity which may be known in Numbers, by determining the Length of the Pendulum at two different Latitudes. The other Member of the Equation is an Expression of the same Quantity, as it is deduced by the preceding Calculus. Secondly, This new Equation B cannot be of any Service in determining the Coefficients and Exponents \(f, g, p, q, \&c.\). For we shall now shew, that the foregoing Ratio \(p-p\) has such an immediate Connexion with \(\alpha\), that one of them being determin'd, the other will necessarily be so too, independently of \(Pp\) the Values of \( f, g, p, q, \&c. \) This may deserve our Attention, and the Proof is thus: XXII. Because the Ratio of the Gravity to the Centrifugal Force is very great, and is express'd by \( m \), in the Equation A we may reject the third and fourth Terms; by which means the Equation will be reduced to this, \[ \frac{f}{3+p} + \frac{g}{3+q} = \frac{4mfa}{3+p \times 5+p} + \frac{4mgq}{3+q \times 5+q}. \] And if from this Equation we deduce the Value either of \( f \) or \( g \), and substitute it in the Equation B; (having first rejected the first and fourth Terms of the Denominator, as in this Case may be done) we shall have after the Calculation is made, whatever is the Number of Terms in the Equation of the Densities, \[ \frac{p-p}{p} = \frac{10}{4m} - \alpha, \quad \text{or} \quad \frac{p-p}{p} = \frac{1}{115} - \alpha, \quad \text{by putting } 288 \text{ for } m, \] as has been long known. It is easily seen from this Equation, that when \( \alpha \) is determined, \( \frac{p-p}{p} \) will be so too, which was the thing proposed to be proved. XXIII. But from this Equation there follows a very singular Proposition, and which, in some sort, is contrary to the Sentiments of Sir Isaac Newton, Page 430. of the 3d Edition of his Principles. And this is, that if by Observation it shall be discover'd, that the Earth is flatter than according to the Spheroid of Sir Isaac, that is, if the Diameter of the Equator exceeds the Axis by more than the \( \frac{1}{230} \) Part, the Gravity Gravity will increase less from the Equator towards the Pole, than according to the Table which he has given for his Spheroid; Prop. XX. of the 3d Book. And on the contrary, if the Spheroid is not so flat, the Gravity will increase more from the Equator towards the Pole. XXIV. 'Tis thus that Sir Isaac Newton expresses himself about it, when he relates the Experiments made towards the South, concerning the Diminution of Gravity, which Experiments make it greater than his Theory requires. He affirms, that the Earth is denser towards the Centre than at the Superficies, and more depress'd than his Spheroid requires. But by the foregoing Theory we may easily perceive, that if the Density of the Earth diminishes from the Centre towards the Superficies, the Diminution of Gravity from the Pole towards the Equator will be greater than according to Sir Isaac's Table; but at the same time the Earth will be not so much depress'd as his Spheroid requires, instead of being more so, as he affirms. Yet I would not by any means be understood to decide against Sir Isaac's Determination, because I cannot be assured of his Meaning, when he tells us, that the Density of the Earth diminishes from the Centre towards the Circumference. He does not explain this, and perhaps instead of the Earth's being compos'd of parallel Beds or Strata, its Parts may be conceived to be otherwise arranged and disposed, so as that the Proposition of Sir Isaac shall be agreeable to the Truth. XXV. As to Dr. Gregory, who has attempted to comment upon this Passage of Sir Isaac, I think I have demonstrated, that he has committed a Paralogism. He says (Element. Astronom. Lib. III. §. VIII. Prop. LII. Schol.) that if the Earth is denser towards the Centre, or if (for Example) it has a Nucleus of greater Weight than the other Parts, the Diminution of Gravity from the Pole towards the Equator shall be greater than if the whole were of the same Density; and in this he is right. But he is in the wrong (I think) immediately to conclude from thence, that the Earth has a greater Flatness. Whence can he conclude this? It can be only from that Proposition of Sir Isaac which informs us, that Gravity is in a reciprocal Ratio of the Distances; because he gave us the Proposition but the Page before, as a Method for determining the Figure of the Earth. But we are not allow'd to make use of this Proposition in this Case, because it has been shewn, Art. XVIII. that it can take Place only on the Supposition of a homogeneous Spheroid. Therefore, &c. XXVI. It will not be very difficult, without any Regard had to the foregoing Theory, to find the Ratio of the Axes of a Spheroid, which we may suppose to have a Nucleus at the Centre, of greater Density than the rest of the Planet; and hence we shall be easily assured of Dr. Gregory's Mistake. XXVII. Setting aside all Attraction of the Parts of Matter, if the Action of Gravity is directed towards a Centre, and is in the reciprocal Ratio of the Squares of of the Distances, the Ratio of the Axes of the Spheroid will then be that of $\frac{576}{577}$: And the Gravity at the Pole is greater than at the Equator by $\frac{1}{44}$th Part, or thereabouts. Which may be a Confirmation of what is here advanced, especially to such as will not be at the Pains of going through the foregoing Calculations. For we may consider the Spheroid now mention'd, in which Gravity acts in a reciprocal Ratio of the Squares of the Distances, as composed of Matter of such Rarity, in respect of that at the Centre, that the Gravity is produced only by the Attraction of the Centre or Nucleus. XXVIII. In the foregoing Calculations, in order to find the Axes of our Spheroids, and to know whether their Figure makes a sensible Approach to that of the conical Ellipsis, we have had Recourse to this Principle, that Gravity ought always to act in a Direction perpendicular to the Surface. Two Reasons have prevail'd with us to make use of this Principle rather than the other, which consists in the Equilibrium of the Columns. The first is, because the Calculations founded thereon are more simple. The second is, that considering the State of the actual Solidity of the Earth, it should seem as if this Principle were the more indispensably necessary. However, because Sir Isaac Newton, and all the other Philosophers, who have treated about the Figure of the Earth, have taken it, as it were, at its first Formation, at which Time they suppose it to have been fluid; we shall here make the same Supposition, and we shall assume no other Ratio for that of the two Axes, than that of the Spheroid, which results from a Coincidence of these two Principles. We shall begin by inquiring what is the entire Weight of any Column CN, Fig. 7. To do this we must resume the Expression of the Attraction in any Point M of the Column CN; then multiply it by \( r + \lambda r \), and by the Density \( fr^p + gr^q \), and afterwards we must find the Fluent. Thus we shall have \[ \frac{cf^2e^2 + 2p}{1 + p \times 3 + p} + \frac{cg^2e^2 + 2q}{1 + q \times 3 + q} + \frac{2cfge^2 + p + q}{2 + p + q \times 3 + p} \] \[ + \frac{2cfge^2 + p + q}{2 + p + q \times 3 + q} + \frac{4cf^2ae^2 + 2p}{1 + p \times 3 + p \times 5 + p} \] \[ + \frac{4cg^2ae^2 + 2q}{1 + q \times 3 + q \times 5 + q} + \frac{8cfgae^2 + p + q}{2 + p + q \times 3 + p \times 5 + q} \] \[ + \frac{8cfgae^2 + p + q}{2 + p + q \times 3 + q \times 5 + q} + \frac{4 + 2pcf^2ae^2 + 2p}{1 + p \times 3 + p \times 5 + p} \] \[ + \frac{4 + 2pcg^2ae^2 + q}{3 + q \times 5 + q \times 1 + p} + \frac{8 + 4pcfgae^2 + p + q}{2 + p + q \times 3 + p \times 5 + p} \] \[ + \frac{8 + 4qcfgae^2 + p + q}{2 + p + q \times 3 + q \times 5 + q} \] for the total Gravity of any Column CN, having Regard only to the Attraction. XXIX. If in this Expression we make \( \lambda = 0 \), we shall have the Gravity of the Column at the Pole. XXX. And if we make \( \lambda = \alpha \), we shall have the Aggregate of the Attractions of the Column at the Equator. XXXI. Now because the Column CN is in Equilibrio with the Column CB; it follows from thence, that if we subtract the Weight of the Column CB from the Aggregate of the Attractions of the Column CN, the Residue must be equal to the Sum of the centrifugal Forces. Forces of the Column CN. Now to endue our Spheroids with this Property, we will resume the Expression of the centrifugal Force in E, which we found Art. XIV. which will give \(\left( \frac{8cfe^1 + p\lambda}{3 + p \times 5 + p} + \frac{8cge^1 + p\lambda}{3 + q \times 5 + q} \right)_e\) for that Part of the centrifugal Force which acts according to CM, in any Place M, by expunging the Terms in which \(a\alpha\) would be found. This Value being multiply'd by \(r\), and by the Density, will give (when we have taken the Fluent) \(\frac{8cf^2e^2 + 2p\lambda}{2 + p \times 3 + p \times 5 + p}\) \[ + \frac{8cfg^2e^2 + p + q\lambda}{2 + p \times 3 + q \times 5 + q} + \frac{8cfg^2e^2 + p + q\lambda}{2 + q \times 3 + p \times 5 + p} \] \[ + \frac{8cge^2e^2 + 2q\lambda}{2 + q \times 3 + q \times 5 + q} \] for the Sum of the centrifugal Forces of the Column CN, still expunging those Terms in which either \(a\alpha\) or \(\lambda\lambda\) are found. Then making this Expression equal to \[ \frac{4 + 2pcf^2e^2 + 2p\lambda}{1 + p \times 3 + p \times 5 + p} + \frac{8 + 4pcfge^2 + p + q\lambda}{2 + p + q \times 3 + p \times 5 + p} \] \[ + \frac{8 + 4qcfg^2e^2 + p + q\lambda}{2 + p + q \times 3 + q \times 5 + q} + \frac{4 + 2qce^2e^2 + 2q\lambda}{1 + q \times 3 + q \times 5 + q}, \] which is the Difference of the Weight of the Column at the Pole CB, from the Sum of the Attractions of the Column CN, we shall have the Equation \[ \frac{ppff}{1 + p \times 2 + p \times 3 + p \times 5 + p} + \frac{2pqfg}{2 + p + q \times 3 + p \times 5 + p \times 2 + q} \] \[ + \frac{2pqfg}{2 + p + q \times 3 + q \times 5 + q \times 2 + p} + \frac{qqgg}{1 + q \times 2 + q \times 3 + q \times 5 + q} = 0, \] where we have put \(e = 1\), for the greater Simplicity of Calculation. XXXII. XXXII. This Equation informs us, that when out of all the infinite Varieties, which will be supply'd by the Equation of the Densities $D = fr^p + gr^q + hr^s$, &c. we shall have taken at Pleasure all the Coefficients, and all the Exponents, one only excepted; if this last is such in respect of the others, that it may fulfil the Conditions of the foregoing Equation, the Spheroid, being supposed in a State of Fluidity, will be in Æquilibrio, because it will unite as well the Principle of a perpendicular Tendency to the Surface, as that of an Equipoise of the several Columns. XXXIII. Before I conclude this Paper, I shall make a few Reflections on the Principles we have now made use of, for determining the Figure of a Spheroid revolving about its Axe. The first Principle which, after Mr. Huygens, we have had Recourse to, and which consists in making Bodies gravitate perpendicularly to the Surface, seems to me of absolute Necessity. For if there were never so little Water upon the Surface of the Earth, it could not be at Rest, if it had a Tendency any how inclined to the Surface. The second Principle, made use of by Sir Isaac Newton, and which consists in an Equilibrium of the Columns C E, C N, C P, could be thought necessary (I think) only for these two Reasons: The first is that which is usually assign'd, that at the first Formation of the Earth, it was probably in a State of perfect Fluidity; in which case it must acquire such a Figure, as will result from the Equilibrium of the Columns, and from the Gravitation acting perpendicularly to the Surface. Indeed though this Reason has a Degree of Plausibility, yet there are many who think it to be of small Force. Perhaps, say they, the Earth has never been in this fluid Condition. The second Reason, which I believe will have a greater Weight with every Body, is this. Considering the Earth as it is at present, and without carrying our Thoughts so far back as to its Formation, if the Ocean, which is now upon its Surface, has any considerable Depth, and if its Parts preserve a Communication with each other, from Region to Region, by subterraneous Canals; it can only keep an Equilibrium by this Means, because its Superficies is the same as it would have, were the whole a Fluid. XXXIV. This second Reason has suggested a Reflexion to my Mind, concerning the Equipoise of the Columns now calculated, Art. XXXI. and XXXII. Let us first suppose, that the Earth is our fluid Spheroid, composed of Beds of different Densities; and that afterwards this Fluid hardens into a Solid, so that the different Beds or Strata, of which it is made up, are of no other Use but to cause a Gravity by their Attractions. Then let us suppose, that the Seas and great Waters about the Earth have a Communication with each other, by means of some subterraneous Canals. As the Waters of the Sea, which unite with one another, are probably homogeneous, the foregoing Calculation, wherein we have consider'd the Spheroid as a Fluid, can no longer take Place, because we have there supposed, that the Fluid contain'd in the Canal BCN is of a Density, that varies from the Centre Centre to the Circumference. From hence it seems to me, we must undertake the Computation of the Equilibrium of the Columns after another Manner, thus: We must examine whether two Canals, as CN and BC, which are fill'd with a homogeneous Fluid, will be in Aequilibrio, all the other Parts of the Spheroid continuing as above. XXXV. To do this, we will begin with finding the Gravity of any Column CN, (Fig. 7.) arising from Attraction alone. First, then, we must resume the Expression of the Attraction in any Point M, Art. VII. Then we must multiply it by \( r + \lambda r \), which will give \[ \frac{2cfr^1 + pr}{3 + p} + \frac{8 + 4pcfar^1 + pr}{3 + p \times 5 + p} + \frac{8cfar^1 + pr}{3 + p \times 5 + p} \] \[ + \frac{2cgr^1 + qr}{3 + q}, \text{ &c. And taking the Fluent of this Quantity, we shall have } \] \[ \frac{2cfe^2 + p}{3 + p \times 2 + p} + \frac{4cf \lambda e^2 + p}{3 + p \times 5 + p} \] \[ + \frac{8cf \alpha e^2 + p}{2 + p \times 3 + p \times 5 + p} + \frac{2oge^2 + q}{3 + q \times 2 + q}, \text{ &c. for the Gravity of the whole Column CN. } \] XXXVI. If in this Value we make \( \lambda = 0 \), we shall have the Gravity of the Column at the Pole. XXXVII. And if we subtract the Gravity of the Column at the Pole from the whole Sum of the Attractions of the Column CN, we shall have \[ \frac{4cf \lambda e^2 + p}{3 + p \times 5 + p} + \frac{4cge^2 + q}{3 + q \times 5 + q} \] which must be equal to the Sum of the centrifugal Forces of the Column CN, in in order that the Columns CB and CN may be in Equilibrio. But we shall find this really to obtain, if we resume the Quantity \(\frac{8cfe^1 + p}{3 + p \times 5 + p} + \frac{8cge^1 + q}{3 + q \times 5 + q}\), which expresses (Art. XXXI.) that Part of the centrifugal Force in M, which acts according to CM. Then multiplying this Expression by \(i\), and seeking the Fluent, we shall have \(\frac{4cfe^2 + p}{3 + p \times 5 + p} + \frac{4cge^2 + q}{3 + q \times 5 + q}\) for the Aggregate of the centrifugal Forces of the Column CN. And this being the same as the foregoing, shews, that the Columns CB and CN are in Equilibrio, supposing them to be homogeneous; nor are we here obliged, as in Art. XXXII. where we consider them as heterogeneous, to suppose the Coefficients f, p, &c. to have any certain Relation among one another. XXXVIII. Perhaps it may be urged, that the foregoing Calculus agrees only to a Canal, as BCN, which passes through the Centre; and that we ought to prove, in the same manner, that the Water included in any other Canal pqr would observe an Equilibrium. But it appears to me, that this Property may be derived from the former: For it follows from the foregoing Calculation, that if we might be allow'd to make this Hypothesis, viz. That independently of the Attraction of any Matter, the Gravity at any Distance CN from the Centre, (see Fig. 7.) would be proportional to \(\frac{2cfe^1 + p}{3 + p} + \frac{2p - 2cfae^1 + p}{3 + p \times 5 + p} + \frac{8cfae^1 + p}{3 + p \times 5 + p}\), &c. it is plain from thence, that a Mass of the homogeneous geneous Fluid, which should turn about the Axis C B, would assume the same Form as that of our heterogeneous Fluids. But if this Spheroid should then put on a fix'd State, except only some Canal p q r, the Water in this Canal would be in Equilibrio; for without this, the Spheroid could not be esteem'd as having arrived to its fix'd State. But this Supposition comes to the same as that of our heterogeneous Spheroid, composed of elliptical Beds, in which should be found a Canal p q r of a homogeneous Fluid; provided that the Space, which this Canal possesses in the Globe, be not of so large an Extent, as to change the Law of Attraction. It might now be thought seasonable to give Examples, for illustrating the foregoing Theory; but they are so easy to be produced, after what is already done, that I shall leave them to the inquisitive Reader, having perhaps exceeded the Limits, within which this Discourse should be confined. Therefore I shall only add the few Observations following. The only three Planets, in which we can be assured of Gravitation, and the centrifugal Force, are the Sun, Jupiter, and the Earth. As to the Sun, the centrifugal Force is there so small, in respect of its Gravity, that his Poles must be very little depress'd, so that we cannot be sensible of it by Observation. Then as to Jupiter, Observations make him something less flat than according to Sir Isaac Newton; that is to say, than if he were composed of Matter of an uniform Density. Therefore by the foregoing Theory, he must be a little more dense towards the Centre, than at the Parts near the Superficies. We might make a thousand Hypotheses about the Manner of distributing the the Inequality of Density, proceeding from the Centre towards the Circumference, which would all agree with the Figure observed, and which are very easy to calculate by the Principles here laid down. As to what concerns the Earth, I shall wait till we receive the Observations which must have been lately made in Peru; that by comparing those with what Observations we have made under the arctick Circle, and with those of Mr. Picart in France, we may have the true Difference of the Earth's Diameters at the Equator and at the Poles. Then our Theory may be apply'd, to determine whether the Earth is more or less dense at the central Parts than at the Surface, or whether it be every-where of an uniform Density, as it ought to be, if (without admitting very great Errors in the Observations) it may be concluded, that the Earth is really the Spheroid of Sir Isaac Newton; and this Case would be the simplest and the most natural of all. I am here obliged to acknowledge, that if the Observations we have made in the North may be rely'd upon, and if we must admit as incontestable as well the Measure of a Degree as the Length of the Pendulum, the foregoing Theory could not be reconciled to the Phenomena. For it follows from our Observations, that the Diameter of the Equator must exceed the Earth's Axis by more than $\frac{1}{30}$ Part: And that the Gravity at the Pole must be greater than that at the Equator by more than $\frac{1}{30}$ Part likewise; which will by no means agree with what we have deduced in Art. XXIII. As to what concerns the Measure of Gravity in Lapland, as being not so liable to Error as the measuring During a Degree; the Earth may be not quite so flat as Sir Isaac's Spheroid requires. By the Table of the Length of the Pendulum, exhibited in the Treatise concerning the Figure of the Earth, publish'd this Year by Mr. de Maupertuis, and by Art. XXII. of the present Discourse, the Earth may be more elevated at the Equator than at the Pole by the $\frac{1}{266}$ Part, or thereabouts. After the true Quantity of the Earth's flatness shall be fully settled, if it should be found to have this Figure, I should be apt to think it is a little more dense at the Centre than towards the Superficies. But if on the contrary we should be well ascertain'd, that the Earth is raised higher at the Equator than at the Pole, by above the $\frac{1}{235}$ Part; and if, for any sufficient Reason, we may something shorten the Length of the Pendulum that beats Seconds in the North; there would be some grounds to allow, that the Earth is not so dense at the central Regions as at those near the Surface. But if it shall happen, that we can neither diminish the Length of the Pendulum, nor the Excess of the equatorial Diameter above the Axe; I must then give up my Hypothesis. Yet I shall think it may be of some Use to have thus discus'd it, because possibly no one would have imagined what might have been the Result of it. It appears that even Sir Isaac Newton was of Opinion, that it was necessary the Earth should be more dense towards the Centre, in order to be so much the flatter at the Poles: And that it follow'd from this greater Flatness, that Gravity increased so much the more from the Equator towards the Pole. II. New