Principia Mathematica (Motte Translation, 1848) — part 29 of 45

For if any small part of the vortex, whose particles or physical points preserve a given situation among each other, be supposed to be congealed, this particle will move according to the same law as before, since no change is made either in its density, vis insita, or figure. And again ; if a congealed or solid part of the vortex be of the same density with the rest of the vortex, and be resolved into a fluid, this will move according to the same law as before, except in so far as its particles, now become fluid, may be moved among themselves. Neglect, therefore, the motion of the particles among themselves as not at all concerning the progressive motion of the whole, and the motion of the whole will be the same as before. But this motion will be the same with the motion of other parts of the vortex at equal distances from the centre ; because the solid, now resolved into a fluid, is become perfectly like to the other parts of the vortex. Therefore a solid, if it be of the same density with the matter of the vortex, will move with the same motion as the parts thereof, being relatively at rest in the matter that sur- rounds it. If it be more dense, it will endeavour more than before to re- cede from the centre ; and therefore overcoming that force of the vortex, by which, being, as it were, kept in equilibrio, it was retained in its orbit, it will recede from the centre, and in its revolution describe a spiral, re- turning no longer into the same orbit. And, by the same argument, if it be more rare, it will approach to the centre. Therefore it can never con- tinually go round in the same orbit, unless it be of the same density with the fluid. But we have shewn in that case that it would revolve accord- ing to the same law with those parts of the fluid that are at the same or equal distances from the centre of the vortex.

Cor. 1. Therefore a solid revolving in a vortex, and continually goino- round in the same orbit, is relatively quiescent in the fluid that carries it. Cor. 2. And if the vortex be of an uniform density, the same body may revolve at any distance from the centre of the vortex.

SCHOLIUM.

Hence it is manifest that the planets are not carried round in corporeal vortices ; for, according to the Copernican hypothesis, the planets going

Sec. IX.]

OF NATURAL PHILOSOPHY.

379

round the sun revolve in ellipses, having the sun in their common focus and by radii drawn to the sun describe areas proportional to the times. But now the parts of a vortex can never re- volve with such a motion. Let AD, BE, CF, represent three orbits describ- ed about the sun S, of which let the utmost circle CF be concentric to the sun ; and let the aphelia of the two in- nermost be A, B ; and their perihelia D, E. Therefore a body revolving in the orb CF, describing, by a radius drawn to the sun, areas proportional to the times, will move with an uniform motion. And, according to the laws of astronomy, the body revolving in the orb BE will move slower in its aphelion B, and swifter in its perihelion E ; whereas, according to the laws of mechanics, the matter of the vortex ought to move more swiftly in the narrow space between A and C than in the wide space between D and F ; that is, more swiftly in the aphelion than in the perihelion. Now these two conclusions contradict each other. So at the beginning of the sign of Virgo, where the aphelion of Mars is at present, the distance between the orbits of Mars and Venus is to the distance between the same orbits, at the beginning of the sign of Pisces, as about 3 to 2 ; and therefore the matter of the vortex between those orbits ought to be swifter at the beginning of Pisces than at the beginning of Virgo in the ratio of 3 to 2 ; for the nar- rower the space is through which the same quantity of matter passes in the game time of one revolution, the greater will be the velocity with which it passes through it. Therefore if the earth being relatively at rest in this celestial matter should be carried round by it, and revolve together with it about the sun, the velocity of the earth at the beginning of Pisces would be to its velocity at the beginning of Virgo in a sesquialteral ratio. Therefore the sun's apparent diurnal motion at the beginning of Virgo ought to be above 70 minutes, and at the beginning of Pisces less than 48 minutes; whereas, on the contrary, that apparent motion of the sun is really greater at the beginning of Pisces than at the beginning of Virgo, as experience testifies ; and therefore the earth is swifter at the beginning of Virgo than at the beginning of Pisces ; so that the hypothesis of vor- tices is utterly irreconcileable with astronomical phsenomena, and rather serves to perplex than explain the heavenly motions. How these mo- tions are performed in free spaces without vortices, may be understood by the first Book ; and I shall now more fully treat of it in the following Book.

BOOKIII.

BOOK III.

In the preceding Books I have laid down the principles of philosophy ; principles not philosophical, but mathematical ; such, to wit, as we may build our reasonings upon in philosophical inquiries. These principles are the laws and conditions of certain motions, and powers or forces, which chiefly have respect to philosophy ; but, lest they should have appeared of themselves dry and barren, I have illustrated them here and there with some philosophical scholiums, giving an account of such things as are of more general nature, and which philosophy seems chiefly to be founded on ; such as the density and the resistance of bodies, spaces void of all bodies, and the motion of light and sounds. It remains that, from the same prin- ciples, I now demonstrate the frame of the System of the World. Upon this subject I had, indeed, composed the third Book in a popular method, that it might be read by many ; but afterward, considering that such as had not sufficiently entered into the principles could not easily discern the strength of the consequences, nor lay aside the prejudices to which they had been many years accustomed, therefore, to prevent the disputes which might be raised upon such accounts, I chose to reduce the substance of this Book into the form of Propositions (in the mathematical way), which should be read by those only who had first made themselves masters of the principles established in the preceding Books: not that I would advise any one to the previous study of every Proposition of those Books ; for they abound with such as might cost too much time, even to readers of good mathematical learning. It is enough if one carefully reads the Definitions, the Laws of Motion, and the first three Sections of the first Book. He may then pass on to this Book, and consult such of the remaining Propositions of the first two Books, as the references in this, and his occasions; shall require.

384 , THE MATHEMATICAL PRINCIPLES [BoOK III.

RULES OF REASONING IN PHILOSOPHY.

RULE L

We are to admit no more causes of natural thing's than such as are both true and sufficient to explain their appearances. To this purpose the philosophers say that Nature does nothing in vain, and more is in vain when less will serve ; for Nature is pleased with sim- plicity, and aifects not the pomp of superfluous causes.

RULE IL

Therefore to the same natural effects we Tnust, as far as possible, assign

the same causes. As to respiration in a man and in a heast ; the descent of stones in Europe and in America ; the light of our culinary fire and of the sun ; the reflec- tion of light in the earth, and in the planets.

RULE m.

The qualities of bodies, lohich admit neither intension nor remissioji of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.

For since the qualities of bodies are only known to us by experiments, we are to hold for universal all such as universally agree with experiments ; and such as are not liable to diminution can never be quite taken away. We are certainly not to relinquish the evidence of experiments for the sake of dreams and vain fictions of our own devising ; nor are we to recede from the analogy of Nature, which uses to be simple, and always consonant to itself. We no other way know the extension of bodies than by our senses, nor do these reaoh it in all bodies ; but because we perceive extension in all that are sensible, therefore we ascribe it universally to all others also. That abundance of bodies are hard, we learn by experience ; and because the hardness of the whole arises from the hardness of the parts, we therefore justly infer the hardness of the undivided particles not only of the bodies we feel but of all others. That all bodies are impenetrable, Ave gather not from reason, but from sensation. The bodies which we handle we find im- penetrable, and thence conclude impenetrability to be an universal property of all bodies whatsoever. That all bodies are moveable, and endowed with certain powers (which we call the vires inertim) of persevering in their mo- tion, or in their rest, we only infer from the like properties observed in the

Book III.] of natural philosophy. 385

bodies which we have seen. The extension, hardness, impenetrability, mo- bility, and vis inerticE of the whole, result from the extension, hardness, impenetrability, mobility, and vires inerticB of the parts ; and thence we conclude the least particles of all bodies to be also all extended, and hard, and impenetrable, and moveable, and endowed with their proper vires inertice. And this is the foundation of all philosophy. Moreover, that the divided but contiguous particles of bodies may be separated from one another, is matter of observation ; and, in the particles that remain undivided, our minds are able to distinguish yet lesser parts, as is mathematically demon- strated. But whether the parts so distinguished, and not yet divided, may, by the powers of Nature, be actually divided and separated from one an- other, we cannot certainly determine. Yet, had we the proof of but one experiment that any undivided particle, in breaking a hard and solid body, suffered a division, we might by virtue of this rule conclude that the un- divided as well as the divided particles may be divided and actually sep- arated to infinity.

Lastly, if it universally appears, by experiments and astronomical obser- vations, that all bodies about the earth gravitate towards the earth, and that in proportion to the quantity of matter which they severally contain : that the moon likewise, according to the quantity of its matter, gravitates towards the earth ; that, on the other hand, our sea gravitates towards the moon ; and all the planets mutually one towards another ; and the comets in like manner towards the sun ; we must, in consequence of this rule, uni- versally allow that all bodies whatsoever are endowed with a principle of mutual gravitation. For the argument from the appearances concludes with more force for the universal gravitation of all bodies than for their impen- etrability ; of which, among those in the celestial regions, we have no ex- periments, nor any manner of observation. Not that I affirm gravity to be essential to bodies : by their vis insita I mean nothing but their vis inertice. This is immutable. Their gravity is diminished as they recede from the earth.

RULE IV.

In experimental philosophy we are to look upon propositions collected by genhral induction from phcenomena as accurately or very nearly true, notiuithstanding any contrary hypotheses that 'may be imagined, till such time as other phcenomena occur, by which they may either be m^ade w.ore accurate, or liable to exceptions. This rule we must follow, that the argument of induction may not be-

evaded by hypotheses.

25

386

THE MATHEMATICAL PRINCIPLES

[Book IIL

PHENOMENA, OR APPEARANCES.

PHiENOMENON I.

That the circumjovicd planets, by radii drawn to Jwpitefs centre^ de- scribe areas proportional to the times of description ; and that their periodic times, the fixed stars being at rest, are in tlie sesquiplicate proportion of their distances from, its centre.

This we know from astronomical observations. For the orbits of these planets differ but insensibly from circles concentric to Jupiter ; and their motions in those circles are found to be uniform. And all astronomers agree that their periodic times are in the sesquiplicate proportion of the semi-diameters of their orbits ; and so it manifestly appears from the fol- lowing table.

The periodic times of the satellites of Jupiter.

l\ \SK 27'. 34". 3^. 13h. 13' 42". 7\ 3\ 42' 36". 16'^. 16^ 32' 9".

The distances of the satellites from Jupitefs centre.

From the observations of Borelli

Townly hy the Microm. . . . Cassini by the Telescope . . . Cassini by the eclip. of the satel. .

1 1 2

3

4

51

5,52

5

51

81 8,78 8 9

14

13,47

241 1

24,72 1 semi-diameter of

23 f Jupiter.

25A

From the periodic times

5,667

9,017

14,384

25,299

Mr. Pound has determined, by the help of excellent micrometers, the diameters of Jupiter and the elongation of its satellites after the following manner. The greatest heliocentric elongation of the fourth satellite from Jupiter's centre was taken with a micrometer in a 15 feet telescope, and at the mean distance of Jupiter from the earth was found about 8' 16". The elongation of the third satellite was taken with a micrometer in a telescope of 123 feet, and at the same distance of Jupiter from the earth was found 4' 42". The greatest elongations of the other satellites, at the same dis^ tance of Jupiter from the earth, are found from the periodic times to be 2' 56" 47"', and 1' 51" 6"'.

The diameter of Jupiter taken with the micrometer in a 123 feet tele- scope several times, and reduced to Jupiter's mean distance from the earth, proved always less than 40", never less than 38", generally 39". This di- ameter in shorter telescopes is 40", or 41" ; for Jupiter's light is a little dilated by the unequal refrangibility of the rays, and this dilatation bears a less ratio to the diameter of Jupiter in the longer and more perfect tele- escopes than in those which are shorter and less perfect. The times in

Book. III.] of natural philosophy. 387'

which two satellites, the first and the third, passed over Jupiter's body, were observed, from the beginning of the ingress to the beginning of the egress, and from the complete ingress to the complete egress, with the long tele- scope. And from the transit of the first satellite, the diameter of Jupiter at its mean distance from the earth came forth 37-|-", and from the transit of the third 37 f. There was observed also the time in which the shadow of the first satellite passed over Jupiter's body, and thence the diameter of Jupiter at its mean distance from the earth came out about 37". Let us suppose its diameter to be 37}" very nearly, and then the greatest elonga- tions of the first, second, third, and fourth satellite will be respectively equal to 5,965, 9,494, 15,141, and 26,63 semi-diameters of Jupiter.

PHENOMENON II.

That the circumsaturnal planets, hy radii draion to Saturn^s centre, de- scribe areas proportional to the times of description ; and that their periodic times, the fixed stars being at rest, are in the sesquiplicate proportion of their distances from its centre.

For, as Cassini from his own observations has determined, their distan- ces from Saturn's centre and their periodic times are as follow.

The periodic times of the satellites of Saturn. 1^. 2P. 18' 27". 2\ 17\ 41' 22". 4^. 12^ 25' 12". 16\ 22^. 41' 14^

79^. 7\ 48' 00".

The distances of the satellites from Saturn^ s centre, in semi-diameters of

its ring.

From observations l\l. 2i. 3. 8. 24.

From the periodic times . . . 1,93. 2,47. 3,45. 8, 23,35.

The greatest elongation of the fourth satellite from Saturn's centre is commonly determined from the observations to be eight of those semi- diameters very nearly. But the greatest elongation of this satellite from Saturn's centre, when taken with an excellent micrometer in Mr. Hiiy gens' telescope of 123 feet, appeared to be eight semi-diameters and y"^^- of a semi- diameter. And from this observation and the periodic times the distances of the satellites from Saturn's centre in semi-diameters of the ring are 2,1. 2,69. 3,75. 8,7. and 25,35. The diameter of Saturn observed in the same telescope was found to be to the diameter of the ring as 3 to 7 ; and the diameter of the ring. May 28-29, 1719, was found to be 43" ; and thence the diameter of the ring when Saturn is at its mean distance from the earth is 42", and the diameter of Saturn 18". These things appear so in very long and excellent telescopes, because in such telescopes the apparent magnitudes of the heavenly bodies bear a greater proportion to the dilata- tion of light in the extremities of those bodies than in shorter telescopes.

3S8 THE MATHEMATICAL PRINCIPLES [BoOK III.

If we, then, reject all tlie spurious light, the diameter of Saturn will not amount to more than 16".

PHENOMENON III.

That the five primary planets, Mercury, Venus, Mars, Jupiter, and Sat- urn, with their several orbits, encompass the sun. That Mercury and Venus revolve about the sun, is evident from their moon-like appearances. When they shine out with a full face, they are, in respect of us, beyond or above the sun ; when they appear half full, they are about the same height on one side or other of the sun ; when horned, they are below or between us and the sun ; and they are sometimes, when directly under, seen like spots traversing the sun's disk. That Mars sur- rounds the sun, is as plain from its full face when near its conjunction with the sun, and from the gibbous figure which it shews in its quadratures. And the same thing is demonstrable of Jupiter and Saturn, from their ap- pearing full in all situations ; for the shadows of their satellites that appear sometimes upon their disks make it plain that the light they shine with is not their own, but borrowed from the sun.

PHENOMENON IV.

That the fixed stars being at rest, the periodic ti?nes of the five prirnary planets, and {whether of the sun about the earth, or) of the earth about the sun, are in the sesquiplicate pj'oportion of their m^ean distances from the sun.

This proportion, first observed by Kepler, is now received by all astron- omers ; for the periodic times are the same, and the dimensions of the orbits are the same, whether the sun revolves about the earth, or the earth about the sun. And as to the measures of the periodic times, all astronomers are agreed about them. But for the dimensions of the orbits, Kepler and Bid- lialdus, above all others, have determined them from observations with the greatest accuracy ; and the mean distances corresponding to the periodic times difier but insensibly from those which they have assigned, and for the most part fall in between them ; as we may see from the following table.

The periodic times vnth respect to the fixed stars, of the planets and earth

revolving about the sun, in days and decimal parts of a day.

h n $ J ? ?

10759,275. 4332,514. 686,9785. 365,2565. 224,6176. 87,9692.

The mean distances of the planets and of the earth from the sun,

^ U $

According to Kepler 951000. 519650. 152350.

to Bullialdus 954198. 522520. 152350.

" to the periodic times .... 954006. 520096. 152369.

Book III.] of natural philosophy. 389

J ? ^

According to Kepler 100000. 72400. 38806.

" to Bullialdus • . . 100000. 72398. 38585.

to the periodic times 100000. 72333. 38710.

As to Mercury and Yenus, there can be no doubt about their distances from the sun ; for they are determined by the elongations of those planets from the sun ; and for the distances of the superior planets, all dispute is cut oif by the eclipses of the satellites of Jupiter. For by those eclipses the position of the shadow which Jupiter projects is determined ; whence we have the heliocentric longitude of Jupiter. And from its helio- centric and geocentric longitudes compared together, we determine its distance.

PHENOMENON V.

Then the primary plaimts^ by radii draion to the earthy describe areas no wise proportional to the times ; but that the areas which they describe by radii drawn to the sun are proportional to the times of descrip- tion.

For to the earth they appear sometimes direct, sometimes stationary, nay, and sometimes retrograde. But from the sun they are always seen direct, and to proceed with a motion nearly uniform, that is to say, a little swifter in the perihelion and a little slower in the aphelion distances, so as to maintain an equality in the description of the areas. This a noted proposition among astronomers, and particularly demonstrable in Jupiter, from the eclipses of his satellites ; by the help of which eclipses, as we have said, the heliocentric longitudes of that planet, and its distances from the sun, are determined.

PH.^NOMENON YL

That the moon, by a radius drawn to the earths centre^ describes an area

proportional to the time of description.

This we gather from the apparent motion of the moon, compared with its apparent diameter. It is true that the motion of the moon is a little disturbed by the action of the sun : but in laying down these Phaenomena, I neglect those small and inconsiderable errors.

390 THE MATHEMATICAL PRINCIPLES [BoOK III.

PROPOSITIONS.

PROPOSITION I. THEOREM I.

That the forces hy ivhich tJie circiimjovial planets are continually drawn off from rectilinear ^notions, and retained in their proper orbits, tend to Jupitefs centre ; and are reciprocally as the squares of the distances of the places of those planets /?'o??^ that centre. The former part of this Proposition appears from Phren. I, and Prop.

11 or III, Book I ; the latter from Ph^en. I, and Cor. 6, Prop. IV; of the same

Book.

The same thing we are to understand of the planets which encompass

Saturn, by Phaen. II.

PROPOSITION II. THEOREM II.

That the forces hy which the primary planets are continually dravm off from rectilinear motions, and retained in their proper orbits, tend to the sun ; and are reciprocally as the squares of the distances of the places of those planets from the sun^s centre.

The former part of the Proposition is manifest from Phasn. V, and Prop. II, Book I ; the latter from Phjen. IV, and Cor. 6, Prop. IV, of the same Book. But this part of the Proposition is, with great accuracy, de- monstrable from the quiescence of the aphelion points ; for a very small aberration from, the reciproccd duplicate proportion would (by Cor. 1, Prop. XLV, Book I) produce a motion of the apsides sensible enough in every single revolution, and in many of them enormously great.

PROPOSITION III. THEOREM III.

That the force by lohich the moon is retained in, its orbit tends to the earth ; and is reciprocally as the square of the distance of its place from the eartKs centre.

The former pb.rt of the Proposition is evident from Phsen. VI, and Prop. II or III, Book I ; the latter from the very slow motion of the moon's apo- gee ; which in every single revolution amounting but to 3° 3' in conse- quentia, may be neglected. For (by Cor. 1, Prop. XLV, Book I) it ap- pears, that, if the distance of the moon from the earth's centre is to the semi- diameter of the earth as D to 1, the force, from which such a motion will result, is reciprocally as T>-^^^, i. e., reciprocally as the power of D, whose exponent is 22^3 ; that is to say, in the proportion of the distance something greater than reciprocally duplicate, but which comes 59| times nearer to the duplicate than to the triplicate proportion. But in regard that this motion is owing to the action of the sun (as we shall afterwards

Book III.] of natural philosophy. 391

shew), it is here to be neglected. The action of the sun, attracting the moon from the earth, is nearly as the moon's distance from the earth ; and therefore (by what we have shewed in Cor. 2, Prop. XLV, Book I) is to the centripetal force of the moon as 2 to 357,45, or nearly so ; that is, as 1 to I78f f . And if we neglect so inconsiderable a force of the sun, the re- maining force, by which the moon is retained in its orb^ will be recipro- cally as D^. This will yet more fully appear from comparing this force with the force of gravity, as is done in the next Proposition.

Cor. If we augment the mean centripetal force by which the moon is retained in its orb, first in the proportion of 177|f to 17Sf|-, and then in the duplicate proportion of the semi-diameter of the earth to the mean dis- tance of the centres of the moon and earth, we shall have the centripetal force of the moon at the surface of the earth ; supposing this force, in de- scending to the earth's surface, continually to increase in the reciprocal 'duplicate proportion of the height.

' PROPOSITION lY. THEOREM IV.

That the moon gravitates towards the earth, and by the force of gravity is continually drawn off from a rectilinear ?notion, and retained in its orbit.

The mean distance of the moon from the earth in the syzygies in semi- diameters of the earth, is, according to Ptolemy and most astronomers, 59 : according to Vendelin and Huygens, 60 ; to Copernicus, 60i ; to Street, 60f ; and to Tycho, 56|. But Tycho, and all that follow his ta- bles of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed the refractions of the fixed stars, and that by four or five minutes near the horizon, did thereby increase the moon's horizontal parallax by a like number of minutes, that is, by a twelfth or fifteenth part of the whole parallax. Correct this error, and the distance will become about 60| semi-diameters of the earth, near to what others have assigned. Let us assume the mean distance of 60 diam- eters in the syzygies ; and suppose one revolution of the moon, in respect of the fixed stars, to be completed in 27^^. 7^\ 43', as astronomers have de- termined ; and the circumference of the earth to amount to 123249600 Paris feet, as the French have found by mensuration. And now if we imagine the moon, deprived of all motion, to be let go, so as to descend towards the earth with the impulse of all that force by which (by Cor. Prop. Ill) it is retained in its orb, it will in the space of one minute of time, describe in its fall 15y'2 Paris feet. This we gather by a calculus, founded either upon Prop. XXXVI, Book I, or (which comes to the same thing) upon Cor. 9, Prop. IV, of the same Book. For the versed sine of that arc, which the moon, in the space of one minute of time, would by its mean

392 THE MATHEMATICAL PRINCIPLES [BoOK III.

motion describe at the distance of 60 semi-diameters of the earth, is nearly 15 j\ Paris feet, or more accurately 15 feet, 1 inch, and 1 line f. Where- fore, since that force, in approaching to the earth, increases in the recipro- cal duplicate proportion of the distance, and, upon that account, at the surface of the earth, is 60 X 60 times greater than at the moon, a body in our regions, falling with that force, ought in the space of one minute of time, to describe 60 X 60 X 15j\ Parisieet] and, in the space of one sec- ond of time, to describe 15y\ of those feet; or more accurately 15 feet, 1 inch, and 1 line f . And with this very force we actually find that bodies here upon earth do really descend ; for a pendulum oscillating seconds in the latitude of Paris will be 3 Paris feet, and 8 lines ^ in length, as Mr. Huygens has observed. And the space which a heavy body describes by falling in one second of time is to half the length of this pendulum in the duplicate ratio of the circumference of a circle to its diameter (as Mr. Huygens has also shewn), and is therefore 15 Paris feet, 1 inch, 1 line f . And therefore the force by which the moon is retained in its orbit becomes, at the very surface of the earth, equal to the force of gravity which we ob- serve in heavy bodies there. And therefore (by Rule I and II) the force by which the moon is retained in its orbit is that very same force which we commonly call gravity ; for, were gravity another force different from that, then bodies descending to the earth with the joint impulse of both forces would fall with a double velocity, and in the space of one second of time would describe 30^ Paris feet ; altogether against experience.

This calculus is founded on the hypothesis of the earth's standing still ; for if both earth and moon move about the sun, and at the same time about their common centre of gravity, the distance of the centres of the moon and earth from one another will be 60|- semi-diameters of the earth ; as may be found by a computation from Prop. LX, Book I.

SCHOLIUM.

The demonstration of this Proposition may be more diffusely explained after the following manner. Suppose several moons to revolve about the earth, as in the system of Jupiter or Saturn ; the periodic times of these moons (by the argument of induction) would observe the same law which Kepler found to obtain among the planets ; and therefore their centripetal forces would be reciprocally as the squares of the distances from the centre of the earth, by Prop. I, of this Book. Now if the lowest of these were very small, and were so near the earth as almost to touch the tops of the highest mountains, the centripetal force thereof, retaining it in its orb, would be very nearly equal to the weights of any terrestrial bodies that should be found upon the tops of those mountains, as may be known by the foregoing computation. Therefore if the same little moon should be '.deserted by its centrifugal force that carries it through its orb, and so be

Book III] op natural philosophy. 393

disabled from going onward therein, it would descend to the earth ; and that with the same velocity as heavy bodies do actually fall with upon the tops of those very mountains ; because of the equality of the forces that oblige them both to descend. And if the force by which that lowest moon would descend were diiferent from gravity, and if that moon were to gravi- tate towards the earth, as we find terrestrial bodies do upon the tops of mountains, it would then descend with twice the velocity, as being impel- led by both these forces conspiring together. Therefore since both these forces, that is, the gravity of heavy bodies, and the centripetal forces of the moons, respect the centre of the earth, and are similar and equal between themselves, they will (by Rule I and II) have one and the same cause. And therefore the force which retains the moon in its orbit is that very force which we commonly call gravity ; because otherwise this little moon at the top of a mountain must either be without gravity, or fall twice as swiftly as heavy bodies are wont to do.

PROPOSITION Y. THEOREM V.

That the circumjovial planets gravitate toioards Jupiter ; the circumsat- urnal toivards Satnrn ; the circumsolar toivards the sim ; and by the forces of their gravity are drawn off from rectilinear motions, and re- tained in curvilinear orbits.

For the revolutions of the circumjovial planets about Jupiter, of the circumsaturnal about Saturn, and of Mercury and Venus, and the other circumsolar planets, about the sun, are appearances of the same sort with the revolution of the moon about the earth ; and therefore, by Rule II, must be owing to the same sort of causes; especially since it has been demonstrated, that the forces upon which those revolutions depend tend to the centres of Jupiter, of Saturn, and of the sun ; and that those forces, in receding from Jupiter, from Saturn, and from the sun, decrease in the same proportion, and according to the same law, as the force of gravity does in receding from the earth.

Cor. 1. There is, therefore, a power of gravity tending to all the plan- ets ; for, doubtless, Yenus, Mercury, and the rest, are bodies of the same sort with Jupiter and Saturn. And since all attraction (by Law III) is mutual, Jupiter will therefore gravitate towards all his own satellites, Sat- urn towards his, the earth towards the moon, and the sun towards all the primary planets.

Cor. 2. The force of gravity which tends to any one planet is re- ciprocally as the square of the distance of places from that planet's centre.

Cor. 3. All the planets do mutually gravitate towards one another, by Cor. 1 and 2. And hence it is that Jupiter and Saturn, when near their

394 THE MATHEMATICAL PRINCIPLES [BoOK III.

conjunction; by their mutual attractions sensibly disturb eacb other's mo- tions. So the sun disturbs the motions of the moon ; and both sun and moon disturb our sea, as we shall hereafter explain.

SCHOLIUM.

The force which retains the celestial bodies in their orbits has been hitherto called centripetal force ; but it being now made plain that it can be no other than a gravitating force, we shall hereafter call it gravity. For the cause of that centripetal force which retains the moon in its orbit will extend itself to all the planets, by Rule I, II, and lY.

PROPOSITION YL THEOREM YL

That all bodies gravitate toioards every planet ; and that the weights of bodies toioards any the same planet, at equal distances from the centre of the planet, are proportional to the quantities of Tnatter which they severally contain.