uncertain. Hypothefes were formed on the Nature of primitive Gra- occafioned vity, which gave to the Earth, fuppofed at reft, a Figure whofe Alter- doubts with ation agreed with the Theory of centrifugal Forces, and with the oblong Figure towards the Poles refulting from the actual Measures.

For the Question of the Figure of the Earth depends on the Law according to which primitive Gravity acts, and it is certain, for Example, that if this Force depended on a Cause which would make it draw fometimes to one side and at other Times to another, and which increased or diminished without any conftant Law, neither Theory nor Obfervation ever could determine this Figure.

VII.

regard to the figure of the

Earth.

taken at the

To decide this Queftion finally it was Neceffary to Measure a Degree un- The mea der the Equator, and another within the polar Circle; if the French Af- fures of the tronomers gave Occafion to the Doubts raised concerning the true Figure meridian of the Earth, yet in Juftice to them it must be acknowledged, that it is to their indefatigable Industry we are indebted for the Confirmation of the and at the Theory of Newton, with Refpe&t to the Figure of the Earth, whofe De- equator preffion towards the Poles is now univerfally allowed.

VIII.

polar circle

confirms the theory

of Newton.

fitions made

by Newton

earth.

In determining the Ratio of the Axes of the Earth, Newton befides the mutual Attraction of the Parts of Matter fuppofes the Earth to be an Elliptic Spheroid, and that its Matter is Homogeneous; Maclaurin in his Two fuppo excellent Piece on the Tides which carried the Prife of the royal Academy of Sciences in 1740, was the first who demonftrated that the Earth fup- in determin pofed Fluid and Homogeneous, whose Parts attract each other mutually and ing the fiare befides Attracted by the Sun and Moon, revolving about its Axis, would gure of the neceffarily affume the Form of an Elliptic Spheroid, and demonstrated fur-Maclaurin ther, that in this Spheroid not only the Direction of Gravity was perpendi- verified the cular to the Surface, and the Central Columns in Equilibrio, but that any first. Point whatfoever within the Spheroid was equally preffed on every Side; which last Point was no lefs Neceflary to be proved than the two first, in Order to be affured that the Fluid was in Equilibrio, yet had been neglected by all those who before treated of the Figure of the Earth.

feoond is

falfe,

The Cafe is not the fame with regard to the fecond Suppofition viz. It is proba the Homogeneity of the Matter of the Earth, for it is very poffible ble that the (and Newton himself was of Opinion Prop. 20 B. 3) that the Density of the Earth increases in approaching the Center, now, the different Denfities of the Strata of Matter compofing the Earth fhould change the Law according to which the Bodies of which it is compofed Gravitate, and of Confequence fhould alter the Proportion of its Axes.

of the axes

IX.

Clairaut improving on the Researches of Maclaurin has fhewn that aThe ratio mong all the most probable Hypothefes that can be framed concerning the of the earth Denfity of the interior Parts of the Earth cofidered as an Elliptic Spheroid, decreases in that adopting Attraction, there always fubfifts fuch a Connexion between Proportion the Fraction expreffing the Difference of the Axes, and that which exas gravity increases at preffes the Decrease of Gravity from the Pole to the Equator, that if one the poles, of thofe two Fractions exceeds by any Quantity, the other will be exactly so much lefs; fo that fuppofing, for Inftance, that the excess of the equatorial Diameter above the Axe is, a Suppofition conformable with the actual Measures, we shall have or for the Quantity to be fubtracted from in Order to obtain the total Abreviation of the Pendulum in advancing from the Pole to the Equator, that is to fay, that this Abreviation or what comes to the fame the total Diminution of Gravity, will be or nearly.

I

Now, as all the Experiments on Pendulums fhew that the Diminution of Gravity from the Pole to the Equator, far from being less than as this Theory requires, is much greater, it follows, that the actual Meafures in this Point are inconfiftant with the Theory of the Earth confidered as an Elliptic Spheroid.

It follows from the Theory of Clairaut, that admitting, the Suppofitions the moft natural we can conceive or imagine with regard to the internal Structure of the Earth confidered as an oblate Elliptic Spheroid, that the Ratio of the Axes cannot exceed that of 229 to 230 fince this Ratio is what arifes from the Suppofition of the Homogeneity of the Earth, and that it refults from this Theory, that in every other Cafe Gravity increafing, the Depreffion towards the Poles is lefs.

Tho' the Earth fuppofed Fluid and Heterogeneous whofe Parts attract each other mutually, aflumes an Elliptic Form confiftent with the Laws of Hydrostaticks, yet it might equally affume an infinite Number of other Forms confiftent with the fame Laws, as Dalambert has demonftrated, and as a Variation in the Form would neceflarily produce one in the Decrease of Gravity from the Pole to the Equator, and confequently in the Ratio of the Axes, it is highly probable that a Figure will be found that will conduct to a Refult fuch as will reconcile Theory with Obfervation. The Recherches of this eminent Mathematician fhall be explained hereafter. Newton having computed the Ratio of the Axes of the Earth, determines the Excels of its Height, at the Equator above its Height at the Poles, in the following Manner. The Semidiameter (b + c) at the Equator being to the Semidiameter (b) at the Poles, as 230 to 229, c= and 2b 458 c. and the Mean Semidiameter according to Picart's incnturation, being 19615800 Paris Feet, or 3923, 16 Miles,

b

229

(reckoning 5000 Feet for a Mile,) 2 X 19615800 2b+c. confequently 459. c. 2 X 19615800 and the Excefs (c) of the Height of the Earth at the Equator, above its Height at the Poles, is 85472 Feet or 17 Miles, and Subftituting in the Equation 2 X 19615800 = 2b + c. for c its Value, there will refult 459b2 X 19615800 X 229, wherefore the Height (b) at the Poles will be 19573064 and the Height (b+c) at the Equator 19658536 Feet.

X.

the different

regions of

After determining the Relation of the Axes of the Earth fuppofed Ho- What are mogeneous, Newton investigates after the following Manner (Prop. 20 B. 3) the weights what Bodies weigh in the different Regions of the Earth. Since he had of bodies in proved that the Polar and Equatorial Columns, were in Equilibrio when their Lengths were to each other as 229 to 230 it follows that if a Body (B) be the earth. to another (b) as 229 to 230, and the one (B) be placed at the Pole, and the other (b) at the Equator, the Weight (W) of the Body (B) will be equal to the Weight (w) of the Body (b). but if those two Bodies be placed at the Equator the Weight (W) of the Body (B) will be to the Weight (w) of the Body (b) as 229 to 230, wherefore the Weight [W] of the Body [B] at the Pole will be to the Weight [W] of the fame or of an equal Body at the Equator, as 230 to 229, that is reciprocally as thofe Columns, we fee by the fame reasoning, that on all the Columns of Matter compofing the Spheroid, the Weights of Bodies should be inver!ely as thefe Columns, that is as their Distances from the Center: therefore fuppofing the Distance, of any Place on the Surface of the Earth, from the Center to be known, the Weight of a Body in this Place will be known, and confequently the Quantity of the Increase or Decrease of Gravity, in advancing towards the Poles or the Equator: but as the Distance of any Place from the Center decreases nearly as the Square of the Sine of the Latitude, or as the Verse Sine of double the Latitude as may eafly be proved by Calculation, we fee how Newton formed the Table given (Prop. 20 B. 3) where he lays down the Decrease of Gravity in advancing from the Pole to the Equator.

Example. The Latitude of Paris being 48 50m that of Places under the Equator oo oom and that of Places under the Poles 90d, the verse Sines of double thofe Latitudes are 1134, 00000,and 20000,and the Force of Gravity (g) at the Poles being to the Force of Gravity (G) at the Equator as 230 to 229, the Excefs (g- G or E) of the Force of Gravity at the Pole, is to the Force of Gravity (G) at the Equator as 230 229 to 229, or as I to 229 but the Excels (e) of the Force of Gravity in the Latitude of Paris is to the Excefs (E) of the Force of Gravity at the Poles as 11334 to 20000,wherefore by the Compofition of Ratios, ex E is to EXG, or the Excefs [e] of the Force of Gravity in the Latitude of Paris is to the Force of Gravity [G] at the Equator as IX11334 to 229X20000,

They are

chronal pen

dulums.

that is, as 5667 to 2290000, and the Force of Gravity [e+G] in the Latitude of Paris is to the Force of Gravity [G] at the Equator as 5667+229000 o, that is, as 2295667 to 229000. By a like Calculus the Force of Gravity in any other Latitude is determined.

As Gravity is the fole Caufe of the Ofcillations of Pendulums, the proportional flackning of thefe Ofcillations proves the Diminution of Gravity, and to the len their Acceleration proves that Gravity acts more powerfully; but it is degths of fyn monftrated that the Celerity of the Ofcillations of Pendulums is inverfely as the Length of the Thread to which they are fufpended, therefore when in Order to render the Vibrations of a Pendulum in a certain Latitude synchronal with its Vibrations in another Latitude, it must be fhortened or lengthned, we should conclude that Gravity is lefs or greater in this Region than in the other; Hugbens has determined the Relation which fubfifts between the Quantity a Pendulum is lengthned or fhortened and the Diminution or Augmentation of Gravity; fo that this Quantity being proportional to the Augmentation or Diminution of the Weight, Newton has given in his Table the Length of Pendulums inftead of the Weights.

of latitude

Example. The Length of the Pendulum in the Latitude of Paris being 3f. 81, 561, the Gravity in the Latitude of Paris [2295667] is to the Gravity at the Equator [2290000] as the Length of the Pendulum in the Latitude of Paris [3.8, 561] to the Length of the Pendulum at the Equator [3, 71,684] By a like Calculus the Length of the Pendulum in any other Latitude is determined.

The Degrees of Latitude decreafing in the Spheroid of Newton in the The degrees fame Proportion as the Weights, the fame Table gives the Quantity of are in the the Degrees in Latitude commencing from the Equator where the Latitude fame pro- is od to the Pole where it is 90. portion.

Example. The Length of a Degree [d] at the Poles, being to the Length of a Degree [D] at the Equator, as the Ray of the Circle which has the fame Curviture as the Arc of the Meridian at the Pole, is to the Ray of the Circle which has the fame Curviture as the Arc of the Meridian at the Equator of the Earth, that is, by the Property of the Ellipfis, as the Cube of 230 to the Cube of 229, that is, as 12167000 to 12008989, the Excefs [d-D or E] of the Degree at the Pole is to the Degree [D] at the Equator, as 158011 to 12008989; but the Excefs [e] of a Degree in the Latitude of Paris, is to the Excefs [E] of the Degree at the Pole, as 11334 to 20000 veríe Sines of Double of thofe Latitudes. Wherefore by the Compofition of Ratios exE is to EXD,or the Excefs [e] of a Degree in the Latitude of Paris is to the Length of the Degree [D] at the Equator, as 895448337 is to 12008989000; ând the Length [e+D] of a Degree in the Latitude of Paris is to the Length of a

Degree [D] at the Equator, as 120985338337 to 120089890000; but the Length of a Degree in the Latitude of Paris, according to Picard's, Mensuration is 57061 Toifes, wheretore the Length of a Degree at the Equator is 56637. By a like Calculus the Length of a Degree in any other Latitude

is Determined.

Newton's Table gives the decrease of Gravity from the Pole to the Equator fomewhat less than what refults from actual Measures, but this Table is only calculated for the Case of Homogeneity; and he informs us at the End of