自然哲学的数学原理-42

of all bodies whatsoever. That all bodies are rnoveable, and endowed withcertain powers (which we call the vires inertias] of persevering in their motion, or in their rest, we only infer from the like properties observed in theBOOK 1II.J OF NATURAL PHILOSOPHY. 385bodies which we have seen. The extension, hardness, impenetrability, mobility, and vis inertia of the whole, result from the extension, hardness,impenetrability, mobility, and vires inertia of the parts; and thence weconclude the least particles of all bodies to be also all extended, and hardand impenetrable, and moveable, and endowed with their proper vires inertia.And this is the foundation of all philosophy. Moreover, that the dividedbut contiguous particles of bodies may be separated from one another, ismatter of observation ; and, in the particles that remain undivided, ourminds are able to distinguish yet lesser parts, as is mathematically demonstrated. But whether the parts so distinguished, and not yet divided, may,by the powers of Nature, be actually divided and separated from one another, we cannot certainly determine. Yet, had we the proof of but oneexperiment that any undivided particle, in breaking a hard and solid body,suffered a division, we might by virtue of this rule conclude that the undivided as well as the divided particles may be divided and actually separated to infinity.Lastly, if it universally appears, by experiments and astronomical observations, that all bodies about the earth gravitate towards the earth, andthat in proportion to the quantity of matter which they severally contain ;that the moon likewise, according to the quantity of its matter, gravitatestowards the earth; that, on the other hand, our sea gravitates towards themoon ; and all the planets mutually one towards another ; and the cometsin like manner towards the sun ; we must, in consequence of this rule, universally allow that all bodies whatsoever are endowed with a principle otmutual gravitation. For the argument from the appearances concludes withmore force for the universal gravitation of all bodies than for their impenetrability ; of which, among those in the celestial regions, we have no experiments, nor any manner of observation. Not that I affirm gravity to beessential to bodies : by their vis insita I mean nothing but their vis iiicrticz.This is immutable. Their gravity is diminished as they recede from theearth.RULE IV.In experimental philosophy we are to look upon propositions collected bygeneral induction from, phenomena as accurately or very nearly true,notwithstanding any contrary hypotheses that may be imagined, tillsuch time as other phenomena occur, by which they may either be mademore accurate, or liable to exceptions.This rule we must follow, that the argument of induction may not bfevaded by hypotheses.25386 THE MATHEMATICAL PRINCIPLES [BooK III.PHENOMENA, OR APPEARANCES,PHENOMENON I.That the circumjovial planets, by radii drawn to Jupiter s centre, describe areas proportional to the times of description ; and that theirperiodic times, the fixed stars being at rest, are in the sesquiplicateproportion of their distances from, its centre.This we know from astronomical observations. For the orbits of theseplanets differ but insensibly from circles concentric to Jupiter ; and theirmotions in those circles are found to be uniform. And all astronomersagree that their periodic times are in the sesquiplicate proportion of thesemi-diameters of their orbits; and so it manifestly appears from the fol-1owing table.The periodic times of the satellites of Jupiter. H 18h. 27 . 34". 3d. 13h. 13 42". 7d. 31. 42 36". 16d. 16h. 32 9".The distances of the satellites from Jupiter s centre.Mr. Pound has determined, by the help of excellent micrometers, thediameters of Jupiter and the elongation of its satellites after the followingmanner. The greatest heliocentric elongation of the fourth satellite fromTupiter s centre was taken with a micrometer in a 15 feet telescope, and atthe mean distance of Jupiter from the earth was found about 8 16". Theelongation of the third satellite was taken with a micrometer in a telescopeof 123 feet, and at the same distance of Jupiter from the earth was found4 42". The greatest elongations of the other satellites, at the same distance of Jupiter from the earth, are found from the periodic times to be 256" 47 ", and 1 51" 6 ".The diameter of Jupiter taken with the micrometer in a 123 feet telescope several times, and reduced to Jupiter s mean distance from the earth,proved always less than 40", never less than 38", generally 39". This diameter in shorter telescopes is 40", or 41"; for Jupiter s light is a littledilated by the unequal refrangibility of the rays, and this dilatation bears3 less ratio to the diameter of Jupiter in the longer and more perfect teleescopesthan in those which are shorter and less perfect. The times :iHOOK. III.] OF NATURAL PHILOSOPHY 387which two satellites, the first and the third, passed over Jupiter s body, wereobserved, from the beginning of the ingress to the beginning of the egress,and from the complete ingress to the complete egress, with the long telescope. And from the transit of the first satellite, the diameter of Jupiterat its mean distance from the earth came forth 37 J-". and from the transitof the third 371". There was observed also the time in which the shadowof the first satellite passed over Jupiter s body, and thence the diameter ofJupiter at its mean distance from the earth came out about 37". Let ussuppose its diameter to be 37}" very nearly, and then the greatest elongations of the first, second, third, and fourth satellite will be respectivelyequal to 5,965, 9,494, 15,141, and 26,63 semi-diameters of Jupiter.PHENOMENON II.Tkat the. circumsalurnal planets, by radii drawn, to Saturtfs centre, describe areas proportional to the times of description ; and that theirperiodic times, the fixed stars being at rest, are in the sesqniplicataproportion uf their distances from its centre.For, as Cassiui from his own observations has determined, theii distances from Saturn s centre and their periodic times are as follow.The periodic times of the satellites of Saturn.l d. 2l h. IS 27". 2d. 17h. 41 22". 4d. 12". 25 12". 15d. 22^. 41 14",79 1. 71. 48 00".The distances of the satellitesfrom Saturn s centre, in semi-diameters ojitv ring.From observations li-. 2f 3|. 8. 24From the periodic times . . . 1,93. 2,47. 3,45. 8. 23.35.The greatest elongation of the fourth satellite from Saturn s centre iscommonly determined from the observations to be eight of th-se semidiametersvery nearly. But the greatest elongation of this satellite fromSaturn s centre, when taken with an excellent micrometer iuMr../fuygen8>telescope of 123 feet, appeared to be eight semi-diameters and T7- of a semidiameter.And from this observation arid the periodic times the distancesof the satellites from Saturn s centre in serni-diameters of the ring are 2.1.2,69. 3,75. 8,7. and 25,35. The diameter of Saturn observed in the sametelescope was found to be to the diameter of the ring as 3 to 7 ; and thediameter of the ring, May 28-29, 1719, was found to be 43"; and th:*ncethe diameter of the ring when Saturn is at its mean distance from theearth is 42", and the diameter of Saturn 18". These things appear so invery long and excellent telescopes, because in such telescopes the apparentmagnitudes of the heavenly bodies bear a greater proportion to the dilatation of light in the extremities of those bodies than in shorter telescopes.3S8 THE MATHEMATICAL PRINCIPLES [HoOK IIIIf we, then, reject all the spurious light, the diameter of Saturn will notamount to more than 16".PHENOMENON III.That the five primary planets, Mercury, Venus, Mars, Jupiter, and Saturn, with their several orbits, encompass the sun.That Mercury and Venus revolve about the sun, is evident from theirmoon-like appearances. When they shine out with a full face, they are, inrespect of us, beyond or above the sun ; when they appear half full, theyare 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, whendirectly under, seen like spots traversing the sun s disk. That Mars surrounds the sun, is as plain from its full face when near its conjunction withthe sun. and from the gibbous figure which it shews in its quadratures.And the same thing is demonstrable of Jupiter and Saturn, from their appearing full in all situations ;for the shadows of their satellites that appearsometimes upon their disks make it plain that the light they shine with isnot their own, but borrowed from the sun.PHENOMENON IV.That the fixed stars being at rest, the periodic times of the five primaryplanets, and (whether of the suit about the earth, or) of the earth aboutthe sun, are in the sesquiplicate proportion of their mean distancesfrom the sun.This proportion, first observed by Kepler, is now received by all astronomers ; for the periodic times are the same, and the dimensions of the orbitsare the same, whether the sun revolves about the earth, or the earth aboutthe sun. And as to the measures of the periodic times, all astronomers areagreed about them. But for the dimensions of the orbits, Kepler and Bullialdns,above all others, have determined them from observations with thegreatest accuracy ; and the mean distances corresponding to the periodictimes differ but insensibly from those which they have assigned, and forthe most part fall in between them ; as we may see from the following table.The periodic times with respect to the fixed stars, of the planets and earthrevolving about the sun. in days and decimal parts of a day.* ^ * $ ? *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.* V IAccording to Kepler 951000. 519650. 152350.to Bullialdus 954198. 522520. 152350.to the periodic times .... 954006. 520096. 152369BOOK III.] OF NATURAL PHILOSOPHY. 389J ? *According to Kepler 100000. 72400. 38806" to Bnllialdus ... . . . 100000. 72398. 38585" to the periodic times 100000. 72333. 38710.As to Mercury and Venus, there can be no doubt about their distancesfrom the sun ;for they are determined by the elongations of those planetsfrom the sun ; and for the distances of the superior planets, all dispute iscut off by the eclipses of the satellites of Jupiter. For by those eclipsesthe position of the shadow which Jupiter projects is determined ; whencewe have the heliocentric longitude of Jupiter. And from its heliocentric and geocentric longitudes compared together, we determine itsdistance.PHENOMENON V.Then the primary planets, by radii drawn to the earth, describe areas nowise proportional to the times ; but that the areas which they describeby radii drawn to the snn are proportional to the times of description.For to the earth they appear sometimes direct, sometimes stationary,nay, and sometimes retrograde. But from the sun they are always seendirect, and to proceed with a motion nearly uniform, that is to say, a littleswifter in the perihelion and a little slower in the aphelion distances, so asto maintain an equality in the description of the areas. This a notedproposition among astronomers, and particularly demonstrable in Jupiter,from the eclipses of his satellites; by the help of which eclipses, as we havesaid, the heliocentric longitudes of that planet, and its distances from thesun, are determined.PHENOMENON VI.That the moon, by a radius drawn to the earths centre, describes an areaproportional to the time of description.This we gather from the apparent motion of the moon, compared withits apparent diameter. It is true that the motion of the moon is a littledisturbed by the action of the sun : but in laying down these PhenomenaI neglect those imall and inconsiderable errors.390 THE MATHEMATICAL PRINCIPLES [BOOK IIIPROPOSITIONSPROPOSITIONI. THEOREM I.That the forces by which the circumjovial planets are continually drawnofffrom rectilinear motions, and retained in their proper orbits, tendto Jupiter s centre ; and are reciprocally as the squares of the distancesof the places of those planets/ro?/i that centre.The former part of this Proposition appears from Pham. I, and Prop.II or III, Book I : the latter from Phaen. I, and Cor. 6, Prop. IV, of the sameBook.The same thing we are to understand of the planets which encompassSaturn, by Phaon. II.PROPOSITION II. THEOREM II.That the forces by which the primary planets are continually drawn offfrom rectilinear motions, and retained in their proper orbits, tend tothe sun. ; and are reciprocally as the squares of the distances of theplaces of those planets from the sun s centre.The former part of the Proposition is manifest from Phasn. V, andProp. II, Book I; the latter from Phaen. IV, and Cor. 6, Prop. IV, of thesame Book. But this part of the Proposition is, with great accuracy, demonstrable from the quiescence of the aphelion points ; for a very smallaberration from the reciprocal duplicate proportion would (by Cor. 1, Prop.XLV, Book I) produce a motion of the apsides sensible enough in everysingle revolution, and in many of them enormously great.PROPOSITION III. THEOREM III.That the force by which the moon is retained in its orbit tends to theearth ; and is reciprocally as the square of the distance of itsplac>>,from the earths centre.The former part of the Proposition is evident from Pha3n. VI, and Prop.II or III, Book I; the latter from the very slow motion of the moon s apogee; which in every single revolution amounting but to 3 3 in consequentia,may be neglected. For (by Cor. 1. Prop. XLV, Book I) it appears, that, if the distance of the moon from the earth s centre is to thesemi-diameter of the earth as D to 1, the force, from which such a motionwill result, is reciprocally as D 2^f 3, i. e., reciprocally as the power of D,whose exponent is 2^^ ; that is to say, in the proportion of the distancesomething greater than reciprocally duplicate, but which comes 59f time?nearer to the duplicate than to the triplicate proportion. But in regardthat this motion is owinsr to the action of the sun (as we shall afterwardsBOOK III.] OF NATURAL PHILOSOPHY. 391shew), it is here to be neglected. The action of the sun, attracting themoon from the earth, is nearly as the moon s distance from the earth ; andtherefore (by what we have shewed in Cor. 2, Prop. XLV. Book I) is to thecentripetal force of the moon as 2 to 357,45, or nearly so; that is, as 1 to178 f-. And if we neglect so inconsiderable a force of the sun, the remaining force, by which the moon is retained in its orb, will be reciprocally as D 2. This will yet more fully appear from comparing this forcewith the force of gravity, as is done in the next Proposition.COR. If we augment the mean centripetal force by which the moon isretained in its orb, first in the proportion of 177%$ to 178ff, and then inthe duplicate proportion of the semi-diameter of the earth to the mean distance of the centres of the moon and earth, we shall have the centripetalforce of the moon at the surface of the earth ; supposing this force, in descending to the earth s surface, continually to increase in the reciprocalduplicate proportion of the height.PROPOSITION IV. THEOREM IV.That the moon gravitates towards the earth, and by thejorce oj gravityis continually drawn off from a rectilinear motion, and retained inits orbit.The mean distance of the moon from the earth in the syzygies in semidiametersof the earth, is, according to Ptolemy and most astronomers,59 : according to Vendelin and Huygens, 60 ;to Copernicus, 60 1 ;toStreet, 60| ; and to Tycho, 56|. But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogetheragainst the nature of light) to exceed the refractions of the fixed stars, andthat by four or five minutes near the horizon, did thereby increase themoon s horizontal parallax by a like number of minutes, that is, by atwelfth or fifteenth part of the whole parallax. Correct this error, andthe distance will become about 60^ semi-diameters of the earth, near towhat others have assigned. Let us assume the mean distance of 60 diameters in the syzygies ; and suppose one revolution of the moon, in respectof the fixed stars, to be completed in 27d. 7h. 43 , as astronomers have determined ; and the circumference of the earth to amount to 123249600Paris feet, as the French have found by mensuration. And now if weimagine the moon, deprived of all motion, to be let go, so as to descendtowards 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 15 T^ Paris feet. This we gather by a calculus, foundedeither upon Prop. XXXVI, Book [, 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 mean392 THE MATHEMATICAL PRINCIPLES [BOOK IIImotion describe at the distance of 60 seini-diameters of the earth, is nearly15^ Paris feet, or more accurately 15 feet, 1 inch, and 1 line . Wherefore, since that force, in approaching to the earth, increases in the reciprocal duplicate proportion of the distance, and, upon that account, at thesurface of the earth, is 60 X 60 times greater than at the moon, a bodyin our regions, falling with that force, ought in the space of one minute oftime, to describe 60 X 60 X 15 T] Paris feet; and, in the space of one second of time, to describe 15 ,。 of those feet; or more accurately 15 feet, 1inch, and 1 line f. And with this very force we actually find that bodieshere upon earth do really descend : for a pendulum oscillating seconds inthe latitude of Paris will be 3 Paris feet, and 8 lines 1 in length, as Mr.Hu.y veus has observed. And the space which a heavy body describesby falling in one second of time is to half the length of this pendulum inthe duplicate ratio of the circumference of a circie to its diameter (as Mr.Htiy^ens has also shewn), and is therefore 15 Paris feet, I inch, 1 line J.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 observe in heavy bodies there. And therefore (by Rule I and II) the force bywhich the moon is retained in its orbit is that very same force which wecommonly call gravity ; for, were gravity another force different from that,then bodies descending to the earth with the joint impulse of both forceswould fall with a double velocity, and in the space of one second of timewould 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 abouttheir common centre of gravity, the distance of the centres of the moon andearth from one another will be 6(H semi-diameters of the earth ;as maybe found by a computation from Prop. LX, Book I.SCHOLIUM.The demonstration of this Proposition may be more diffusely explainedafter the following manner. Suppose several moons to revolve about theearth, as in the system of Jupiter or Saturn : the periodic times of thesemoons (by the argument of induction) would observe the same law whichKepler found to obtain among the planets ; and therefore their centripetalforces would be reciprocally as the squares of the distances from the centreof the earth, by Prop. I, of this Book. Now if the lowest of these werevery small, and were so near the earth as almost to touo the tops of thehighest mountains, the centripetal force thereof, retaining it in its orb,would be very nearly equal to the weights of any terrestrial bodies thatshould be found upon the tops of those mountains, as may be known bythe foregoing computation. Therefore if the same little moon should bedeserted by its centrifugal force that carries it through its orb, and so beBOOK 111.] OF NATURAL PHILOSOPHY. 393lisabled from going onward therein, it would descend to the earth ; andthat with the same velocity as heavy bodies do actually fall with upo-n thetops of those very mountains ; because of the equality of the forces thatoblige them both to descend. And if the force by which that lowest moonwould descend were different from gravity, and if that moon were to gravitate towards the earth, as we find terrestrial bodies do upon the tops ofmountains, it would then descend with twice the velocity, as being impelled by both these forces conspiring together. Therefore since both theseforces, that is, the gravity of heavy bodies, and the centripetal forces of themoons, respect the centre of the earth, and are similar and equal betweenthemselves, they will (by Rule I and II) have one and the same cause. Andtherefore the force which retains the moon in its orbit is that very forcewhich we commonly call gravity ; because otherwise this little moon at thetop of a mountain must either be without gravity, or fall twice as swiftlyas heavy bodies are wont to do.PROPOSITION V. THEOREM V.Vhat the circumjovial planets gravitate towards Jupiter ; the circnntsaturnaltowards Saturn ; the circumsolar towards the sun ; and by t/ieforces of their gravity are drawn off from rectilinear motions, and retained in curvilinear orbits.For the revolutions of the circumjovial planets about Jupiter, of thecircumsaturnal about Saturn, and of Mercury and Venus, and the othercircumsolar planets, about the sun, are appearances of the same sort withthe 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 beendemonstrated, that the forces upon which those revolutions depend tend tothe centres of Jupiter, of Saturn, and of the sun ; and that those forces, inreceding from Jupiter, from Saturn, and from the sun, decrease in the sameproportion, and according to the same law, as the force of gravity does inreceding from the earth.COR. 1. There is, therefore, a power of gravity tending to all the planets ; for, doubtless, Venus, Mercury, and the rest, are bodies of the samesort with Jupiter and Saturn. And since all attraction (by Law III) ismutual, Jupiter will therefore gravitate towards all his own satellites, Saturn towards his, the earth towards the moon, and the sun towards all theprimary planets.COR. 2. The force of gravity which tends to any one planet is reciprocally as the square of the distance of places from that planet scentre.