tes explain were mere Hypothefes devoid of Proof, and though Defcartes Explanation tion of the was more Philofophical, it was no le's Fictitious and Imaginary.

the circula

planets in their orbits.

Newton begins with proving in the first Propofition (a), that the Areas defcribed by a Body revolving round an immoveable Center to which it is It is a centripetal 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 hinders the proportional to the Times, that Body is actuated by a Force directed planets 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.

from flying off by the

tangent.

hinders

Newton has also fhewn (Cor. 1. Prop. 2.) that if the Force acting on a Body, urges it to different Points, it would accelerate or retard the Defcription 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 Existance of a centripetal Force drawing them from the Tangent, fo by their not descending in a straight 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 Deicent, 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 muft neceffarily counteract the Force which make them tend to the Sun, and this Force is called the Projectile Force.

Of the centrifugal

IV.

The Effort exerted by the Planets in Confequence of this Force to reforce of the 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 dire&ly 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 Weft to East.

Suppofing the Medium in which the Planets move to be void of all Refiftance, the Confervation of the projectile Motion in the Planets, is accounted for from the Inertia of Matter, and the firft Law of Motion, but its Phyfical Cause, and the Reaton of its Direction are as yet unknown.

vi.

Newton dif

of the square

After having proved that the Planets are retained in their Orbits by a covers the force urging Force directed to the Sun, Newton demonstrates (Prop. 4.) that the centrithe 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-Inv 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 thofe Circles, is in the tances from inverfe Ratio of the Squares of thofe fame Rays, that is of the Distance of the ratio of their periothofe Bodies from the Center: But by the fecond Law of Kepler, which all dic times and the Planets obferve, their periodic Times are in the fefquiplicate Ratio of diftances. . 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 or from the Sun increases, supposing them to revolve in Circles concentric to bits being the Sun.

vit.

First 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 Obfervations, 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 anfwered 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 that they re of the Foci, and this Curve agrees fo exactly with all the Phenomena, that volve in el it is now univerfally 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

has fhewn

Newton de monstrates that in ellip

prove that the centripetal Force which directs the celeftial Bodies in their Courses, 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 demonstrates (Prop. 15.) that the periodic Times in Ellipfes fes the perio are in the fefquiplicate Ratio of their great Axes; that is, that thofe Times dical times are in the fame Proportion in Ellipfes, and Circles whose Diameters are equal fame propor to the great Axes of those Ellipfes.

are in the

tion as in

circles.

This Curve which the Planets defcribe in their Revolution is endued with Confequent 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 also this fame Proportion.

which re

tains the

decreases as

the square of the dif

The centri

proportion

IX.

Newton, not content with examining the Law that makes the Planets detance. fcribe Ellipfes; he enquired further weather in confequence of this Law: 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, !cribe conic for example, (Prop. 10.) that if the Force be as the Distance from the Center, it will make the Body defcribe a conic Section, whofe Center would be the 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 foci. that no other Law could fubfift in our World in its present State.

Sun being placed in

Manner of

X.

Newton afterwards examines (Prop. 17.) the Curve a Body would defcribe determining with a centripetal Force decreasing in the inverfe Ratio of the Square of the the orbit of Distance, fuppofing the Body let go from a given Point, with a Direc pofing the tion and Velocity affumed at Pleasure.

a planet fup

law of cen

tripetal force to be

given.

To folve this Problem, he fets out with the Remark he had made, (Prop. 16.) that the Velocities of Bodies defcribing conic Sections, are in each Point 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.

confequence

It is eafy to conceive that in confequence of other Laws of centripetal What Force different from that of the Square of the Distances Bodies would Curves in describe 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-tripetalforce tum in the Heavenly Spaces; others would make them describe Spirals, &c. fcribed. and Newton in the 42d Propofition, investigates what are the Curves defcribed in all Sorts of Hypothefis of centripetal Forces.

XII.

would be de

Planets in

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 proportithe lower Apfis on the Contrary, the centrifugal Force furpaffes in its on between the centripeturn the centripetal Force, fince in reafcending 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 cause the Body to defcend to the lower, and reafcend to the higher Apfis perpetually.

move is void

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 Density; but he proved in the Scho- the heavenlium of (Propofition 22. B. 2.) that at the Height of two hundred Miles a-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 Refistance 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 scarcely 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 observed, between the Areas and the Times, is a convincing Proof that this Refiftance is actually infenfible.

XIII.

As we have shewn that the Proportionality of the Times and of the Areas which the Planets defcribe 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 decreases in the inverse

The compa

times and

Ratio of the Square of the Distances. If the Planets which perform their Revolutions round the Sun be furrounded by others which revolve round them, and obferving the fame Proportions in their Revolutions, we may conclude that these Satellites are urged by a centripetal Force directed to their Primaries, and that this Force decreafes as that of the Sun in the duplicate Ratio of the Distance.

We can discover only three Planets attended with Satellites, Jupiter, the Earth, and Saturn; we know that the Satellites of those three Planets defcribe around them Areas proportional to the Times, and confequently are urged by a Force tending to thofe Pianets.

XIV.

riton of the Jupiter and Saturn having each feveral Satellites whofe periodic Times periodic and Distances are known, it is eafy to difcover whether the Times of their diftances of Revolution about their Planet, are to their Diftance in the Proportion difcothe fatellites vered by Kepler; and Observations evince that the Satellites of Jupiter and of Saturn Saturn obferve alfo this fecond Law of Kepler in revolving round their Priand Juptier, proves that maries, and of confequence the centripetal Force of Jupiter and of Saturn the centri-decreafe in the Ratio of the Square of the Distances of Bodies from the Fetal force Centre of those Planets.

of thofe pla

nets is allo in the in

XV.

As the Earth is attended only by one Satellite, namely the Moon, it apverfe ratio of pears at firft View difficult to determine the Proportion in which the Force the fquare acts that makes the Moon revolve in her Orbit round the Earth, as in this of the dif- Cafe we have no Term of Comparison.

tances.

force of the

lows the

How New- Newton has found the Means of fupplying this Defect; his Method is as ton dilcove- follows: All Bodies which fall on the Surface of the Earth, describe accordred that the ing to the Progreffion difcovered by Gallileo, Spaces which are as the Squares attractive of the Times of their Descent. We know the mean Distance of the Moon Earth fol- from the Earth which in round Numbers is about 60 Semidiameters of the Earth; and all Bodies near the Surface of the Earth are confidered as equifame propor- diftant from the Centre; therefore if the fame Force produces the Defcent of fion. heavy Bodies, and the Revolution of the Moon in her Orbit; and if this Force decreases in the Ratio of the Square of the Distance, its Action on Bodies near the Surface of the Earth fhould be 3600 Times greater than what it exerts on the Moon, fince the Moon is 60 Times remoter from the Centre of the Earth; we know the Moon's Orbit, because we know at prefent the Measure of the Earth, we know that the Moon describes this Orbit in 27 Days, 7 Hours, 43 Minutes, hence we know the Arc fhe defcribes in one Minute; now by (Cor. 9 Prop. 4.) the Arc defcribed in a given Time by a Body revolving uniformly in a Circle with a given centripetal Force, is a mean Proportional between the Diameter of this Circle and the right Line defcribed in the Body's defcent during that Time.