to us fometimes Parts which were concealed, and conceal: others that were visible.

This Libration of the Moon arifes from her Motion in an Elliptic Orbit, Its caufe. for if the revolved in a circular Orbit, having the Earth for its Center, and turned about her Axis in the Time of her periodic Motion round the Earth, he would in all Pofitions turn the fame Difc exa&ly towards the Earth.

We are ignorant of the Form of the Surface of the Moon, which is on the other Side of her Difc with Refpe&t to us. Some Philofophers have even attempted to explain its Libration, by affigning a conical Figure to that Part of its Surface, which is concealed from us, and who deny her Rotation round her Axis.

The Surface of the Moon is full of Eminences and Cavities, for which reason she reflects on every Side the Light of the Sun, for if her Surface was even and polished like a Mirror, fhe would only reflect to us the Image of the Sun.

The mean Distance of the Moon from the Earth is nearly 60 diameters of the Earth.

Distance of

Semi- the moon

100 to

from the
earth.
Its diameter

The Diameter of the Moon is to the Diameter of the Earth, as 365, its Mafs is to the Mals of the Earth, as 1 to 39, 788 and its Density Its mafs. is to the Denfity of the Earth, as II to 9.

And lastly, a Body which would weigh 3 Pounds at the Surface of the Earth, transferred to the Surface of the Moon would weigh one Pound. All these Proportions are known in the Moon and not in the other Satellites, because this Planet fupplies a peculiar Element, namely her Action on the Sea, which Newton knew how to measure and to employ for determining her Mass, the Method he pursued in this Enquiry will be unfolded in the Sequel.

Theory of the Primary Planets.

I.

In accounting for the celeftial Motions, the firft Phenomenon that occurs to be explained is the perpetual Circulation of the Planets round the Center of their Revolutions.

By the first Law of Nature every Body in Motion perfeveres in that recticlinear Course in which it commenced, therefore that a Planet may be deflected from the straight Line it tends to defcribe inceffantly, it is Neceffary that a Force different from that which makes it tend to defcribe this straight Line fhould inceffantly A&t on it in order to bend its Courfe into a Curve, in the fame Manner as when a Stone is whirled round in a Sling. The Sling inceffantly reftrains the Stone from flying off in the Direction of the Tangent to the Circle it defcribes.

Its denfity.

What bodies

weigh on its furface.

How the

ancient philofophers

To explain this Phenomenon, the Ancients invented their folid Orbs and Defeartes Vortices, but both one and the other of those Explications and Defcar

tes explain were mere Hypothefes devoid of Proof, and though Defcartes Explanation was more Philosophical, it was no less Fictitious and Imaginary.

the circulation of the planets in

their orbits.

tripetal

hirders the

II.

Newton begins with proving in the first Propofition (a), that the Areas described by a Body revolving round an immoveable Center to which it is tea ce continually urged, are proportional to the Times, and reciprocally in the force which Second, that if a Body revolving round a Center defcribes about it Areas proportional to the Times, that Body is actuated by a Force directed to that Center. Since therefore according to Kepler's Discoveries, the Planets defcribe round the Sun Areas proportional to the Times, they are actuated by a centripetal Force, urging them towards the Sun, and retaining them in their Orbits.

planets

from flying off by the

tangent.

hinders

Newton has alfo fhewn (Cor. 1. Prop. 2.) that if the Force acting on a Body, urges it to different Points, it would accelerate or retard the Delcription of the Areas, which would confequently be no longer proportional to the Times: Therefore if the Areas be proportional to the Times, the revolving Body is not only actuated by a centripetal Force, directed to the central Body, but this Force makes it tend to one and the fame Point.

III.

As the Revolutions of the Planets in their Orbits prove the Exiftance of a centripetal Force drawing them from the Tangent, fo by their not defcending in a ftraight Line towards the Center of their Revolution, we may conclude that they are acted upon by another Force different from the And the pro Centripetal. Newton has examined (b) in what Time each Planet would jectile force defcend from its prefent Distance to the Sun if they were actuated by no them from other Force but the Sun's Action, & he has found (P.36) that the different Plafalling to nets would employ in their Defcent, the Half of the periodic Time of the the center Revolution round the Sun of a Body placed at Half their prefent Distances, and confequently thefe Times would be to their periodic Times, as I to 4√2. Thus, Venus for Example would take about 40 Days to defcend to the Sun, for 40: 224: I: 4/2 nearly; Jupiter would employ two Years and a Month in his Defcent, and the Earth and the Moon fixty-fix Days and nineteen Hours, &c. fince then the Planets do not defcend to the Sun, fome Force must neceflarily counteract the Force which make them tend to the Sun, and this Force is called the Projectile Force.

Of the cen

force of the

IV.

The Effort exerted by the Planets in Confequence of this Force to retrifugal cede from the Center of their Motion, is what is called their Centrifugal Force, hence in the Planets, the centrifugal Force is that Part of the projectile Force, which removes them directly from the Center of their Revolution.

planets.

(a) When the Propofitions are quoted without quoting the Book, they are the Propofitions of

the first Book.

(b) De fyftemate mundi, page 31. edition 1731.

V.

The projectile Force has the fame Direction in all the Planets, for they all revolve round the Sun from West to East.

Suppofing the Medium in which the Planets move to be void of all Refiftance, the Conservation of the projectile Motion in the Planets, is ac counted for from the Inertia of Matter, and the first Law of Motion, but its Phyfical Caufe, and the Reafon of its Direction are as yet unknown.

inverse ratio

After having proved that the Planets are retained in their Orbits by a covers the Force directed to the Sun, Newton demonftrates (Prop. 4.) that the centri- force urging the planets petal Forces of Bodies revolving in Circles are to one another as the Squares to the Sun of the Arcs of thofe Circles defcribed in equal Times, divided by their to be in the Rays, from whence he deduces (cor. 6.) that if the periodic Times of Bo- of the fquare dies revolving in Circles be in the fefquiplicate Ratio of their Rays, the cen- of their dif tripetal Force which urges them to the Center of those Circles, is in the tances from inverse Ratio of the Squares of those fame Rays, that is of the Distance of their periothose Bodies from the Center: But by the fecond Law of Kepler, which all dic times and the Planets observe, their periodic Times are in the fefquiplicate Ratio of distances. their Distances from their Center; confequently, the Force which urges fuppofition the Planets towards the Sun, decreases as the Square of their Distance of their orfrom the Sun increases, fuppofing them to revolve in Circles concentric to bits being the Sun.

VII.

the ratio of

Firft in the

circular.

Sun in excen

The first and most natural Notion that we form concerning the Orbits of the Planets, is that they perform their Revolutions in concentric Circles; Before Kepbut the Difference in their apparent Diameters, and more accuracy in the ler's time it wasthought Observations, have long fince made known that their Orbits cannot be con- that the placentric to the Sun; their Courses therefore, before Kepler's Time, were ex- nets revolvplained by excentric Circles, which answered pretty well to the Obfervations ed about the on the Motions of the Sun and the Planets, except Mercury and Mars. tric circles. From confidering the Course of this laft Planet, Kepler fufpe&ed that the But Kepler Orbits of the Planets might poffibly be Ellipfes, having the Sun placed in one has hewn of the Foci, and this Curve agrees fo exactly with all the Phenomena, that volve in el it is now universally acknowledged by Aftronomers, that the Planets revolve lipfes. round the Sun in elliptic Orbits, having the Sun in one of the Foci.

VIII.

Affuming this Discovery, Newton examines what is the Law of centripetal Force, required to make the Planets defcribe an Ellipfe, and he found (Prop. 11.) that this Force must follow the inverse Ratio of the Planet's Distance from the Focus of this Ellipfe. But having found before (cor. 6. Prop. 4.) that if the periodic Times of Bodies revolving in Circles be in the fefquiplicate Ratio of their Rays, the centripetal Forces would be in the inverfe Ratio of those fame Distances; he had no more to do to invincibly

that they re

Newton de monstrates that in ellip

prove that the centripetal Force which directs the celeftial Bodies in their Courfes, follows the inverfe Ratio of the Square of the Distances; but to examine if the periodic Times follow the fame Proportion in Ellipfes as in Circles.

But Newton demonftrates (Prop. 15.) that the periodic Times in Ellipfes fes the perio are in the fefquiplicate Ratio of their great Axes; that is, that those Times are in the are in the fame Proportion in Ellipfes, and Circles whofe Diameters are equal fame propor to the great Axes of those Ellipfes.

dical times

tion as in circles. Confequent

This Curve which the Planets defcribe in their Revolution is endued with this Property, that if fmall Arcs defcribed in equal Times be taken, the ly the centri Space bounded by the Line drawn from one of the Extremities of this Arc, petal force and by the Tangent drawn from the other Extremity increases in the fame Ratio as the Square of the Distance from the Focus decreases; from planets in whence it follows, that the attractive Power which is proportional to this their orbits. Space, follows alio this fame Proportion.

which re tains the

decreases as the fquare of the dif tance.

proportion

IX.

Newton, not content with examining the Law that makes the Planets describe Ellipfes; he enquired further weather in confequence of this Law: The centri Bodies might not defcribe other Curves, and he found (Cor. 1. Prop. 13.) that petal force this Law would only make them describe a conic Section, the Center of the being in this Force being placed in the Focus, let the projectile Force be what it would. Other Laws, by which Bodies might defcribe conic Sections, would make the planets can only de them defcribe them about Points different from the Focus. Newton found, fcribe conic for example, (Prop. 10.) that if the Force be as the Distance from the Center, fections the it will make the Body defcribe a conic Section, whofe Center would be the Sun being placed in Center of Forces, thus Newton has difcovered not only the Law which the one of the centripetal Force obferves in our planetary Syftem, but he has also fhewn that no other Law could fubfift in our World in its present State.

foci.

Manner of

X.

Newton afterwards examines (Prop. 17.) the Curve a Body would describe determining with a centripetal Force decreafing in the inverse Ratio of the Square of the

the orbit of

a planet fup Distance, fuppofing the Body let go from a given Point, with a Direcpofing the tion and Velocity affumed at Pleasure.

law of cen, tripetal

To folve this Problem, he fets out with the Remark he had made, (Prop.

force to be 16.) that the Velocities of Bodies defcribing conic Sections, are in each Point given. of those Curves, as the Square-Roots of the principal Parameters, divided by the Perpendiculars, let fall from the Foci on the Tangents to those Points.

This Propofition is not only very interefting, confidered merely as a geometrical Problem, but also of great ufe in Aftronomy; for finding by Obfervation the Velocity and Direction of a Planet in any Part of its Orbit, by the Affiftance of this Propofition, the Remainder of its Orbit is found out, and the Determination of the Orbits of Comets, may in a great Measure be deduced from this Propofition.

XI.

It is eafy to conceive that in confequence of other Laws of centripetal ¡What Curves in Force different from that of the Square of the Distances Bodies would confequence defcribe other Curves, that there are fome Laws by which notwithftan- of other ding the projectile Force, they would defcend to the Sun, and others by Laws of cen which notwithstanding the centripetal Force, they would recede in infini-tripetal force tum in the Heavenly Spaces; others would make them defcribe Spirals, &c. fcribed. and Newton in the 42d Propofition, inveftigates what are the Curves described in all Sorts of Hypothesis of centripetal Forces.

XII.

would be de

It evidently appears from all that has been faid that the perpetual Circula- The perpetion of the Planets in their Orbits depends on the Proportion between the tual circulacentripetal and the projectile Force, and those who afk why the Planets tion of the arriving at their Perihelia, reafcend to their Aphelia, are ignorant of this their Orbits Proportion; for in the higher Apfis the centripetal Force exceeds the Cen-refults from trifugal Force, fince in defcending the Body approaches the Centre, and in the proporti the lower Apfis on the Contrary, the centrifugal Force furpaffes in its the centripeturn the centripetal Force, fince in reascending the Body recedes from the tal and proCentre: A certain Combination between the centripetal Force and the cen-jectile force. trifugal Force was therefore requifit, that they might alternately prevail and caufe the Body to defcend to the lower, and reafcend to the higher Apfis perpetually.

on between

Another Objection was alledged with regard to the Continuation of the Heavenly Motions, derived from the Refiftance they should undergo in the Medium in which they move. This Objection Newton has anfwered in The medi(Prop. 10. B. 3.) where he fhews that the Refiftance of Mediums diminish um in which in the Ratio of their Weight and their Denfity; but he proved in the Scho- the heavenlium of (Propofition 22. B. 2.) that at the Height of two hundred Miles a-move is void ly Bodies bove the Surface of the Earth, the Air is more rarified than at the Surface, of all refift in the Ratio of 30 to 0,0000000000003998 or nearly as 75000000000000 ance. to 1, from whence he concludes (Prop. 10. B. 3.) fuppofing the Resistance of the Medium in which Jupiter moves to be of this Denfity, this Planet defcribing five of its Semidiameters in 30 days, would from the Refiftance of this Medium, in 1000000 years fcarcely lofe 1000000th Part of its Motion; from hence we fee that the Medium in which the Planets move may be fo rare and fubtile, that its Refiftance may be regarded as Void; and the Proportionality conftantly obferved, between the Areas and the Times, is a convincing Proof that this Refiftance is actually infenfible.

XIII.

As we have fhewn that the Proportionality of the Times and of the Areas which the Planets describe around the Sun, proves that they tend to the Sun as to their Centre, and that the Ratio fubfifting between their periodic Times and their Distances, fhews that this Force decreafes in the invenie