to the centre of the earth are (by Prop. 1, Book 1, Princip. Math.} proportional to the times in which they are described, its velocity, when it returns*o the mountain, will be no less than it was at first; and, retaining the*ame velocity, it will describe the same curve over and over, by the same law514 THE SYSTEM OF THE WORLD.But if we now imagine bodies to be projected in the directions of linesparallel to the horizon from greater heights, as of 5, 10, 100, 1000, or moremiles, or rather as many semi-diameters of the earth, those bodies, according to their different velocity, and the different force of gravity in differentheights, will describe arcs either concentric with the earth, or variouslyeccentric, and go on revolving through the heavens in those trajectories,just as the planets do in their orbs.As when a stone is projected obliquely, that is, any way but in the perpendicular direction, the perpetual deflection thereof towards the earthfrom the right line in which it was projected is a proof of its gravitationto the earth, no less certain than its direct descent when only suffered tofall freely from rest ; so the deviation of bodies moving in free spaces fromrectilinear paths, and perpetual deflection therefrom towards any place, isa sure indication of the existence of some force which from all quartersimpels those bodies towards that place.And as, from the supposed existence of gravity, it necessarily followsthat all bodies about the earth must press downwards, and therefore musteither descend directly to the earth, if they are let fall from rest, or atleast perpetually deviate from right lines towards the earth, if they arcprojected obliquely ;so from the supposed existence of a force directed toany centre, it will follow, by the like necessity, that all bodies upon whichthis force acts mast either descend directly to that centre, or at least deviate perpetually towards it from right lines, if otherwise they should havemoved obliquely in these right lines.And how from the motions given we may infer the forces, or from theforces given we may determine the motions, is shewn in the two first Booksof our Principles of Philosophy.If the earth is supposed to stand still, and the fixed stars to be revolvedin free spaces in the space of 24 hours, it is certain the forces by whichthe fixed stars are retained in their orbs are not directed to the earth, butto the centres of the several orbs, that is, of the several parallel circles,which the fixed stars, declining to one side and the other from the equator,describe daily ;also that by radii drawn to the centres of those orbs thtfixed stars describe areas exactly proportional to the times of description.Then, because the periodic times are equal (by Cor. Ill, Prop. IV, Book 1),it follows that the centripetal forces are as the radii of the several orbs,and that they will perpetually revolve in the same orbs. And the likeconsequences may be drawn from the supposed diurnal motion of theplanets.That forces should be directed to no body on which they physically depend, but to innumerable imaginary points in the axis of the earth, is anhypothesis too incongruous. It is more incongruous still that those forcesshould increase exactly in proportion of the distances from this axis ; forTHE SYSTEM OF THE WORLD. 515this is an indi ation of an increase to immensity, or rather to infinity ;whereas the forces of natural things commonly decrease in receding fromthe fountain from which they flow. But, what is yet more absurd, neitherare the areas described by the same star proportional to the times, nor areits revolutions performed in the same orb ;for as the star recedes from theneighbouring pole, both areas and orb increase; and from the increase ofthe urea it is demonstrated that the forces are not directed to the axis ofthe earth. And this difficulty (Cor. 1, Prop. II) arises from the twofoldmotion that is observed in the fixed stars, one diurnal round the axis ofthe earth, the other exceedingly slow round the axis of the ecliptic. Andthe explication thereof requires a composition of forces so perplexed andso variable, that it is hardly to be reconciled with any physical theory.That there are centripetal forces actually directed to the bodies of thesun, of the earth, and other planets, I thus infer.The moon revolves about our earth, and by radii drawn to its centre(p. 390) describes areas nearly proportional to the times in which they aredescribed, as is evident from its velocity compared with its apparent diameter;for its motion is slower when its diameter is less (and therefore itsdistance greater), and its motion is swifter when its diameter is greater.The revolutions of the satellites of Jupiter about that planet are moreregular (p. 386) : for they describe circles concentric with Jupiter by equable motions, as exactly as our senses can distinguish.And so the satellites of Saturn are revolved about this planet with motions nearly (p. 387) circular and equable, scarcely disturbed by any eccentricity hitherto observed.That Venus and Mercury are revolved about the sun, is demonstrablefrom their moon-like appearances (p. 388) . when they shine with a fullface, they are in those parts of their orbs which in respect of the earth liebeyond the sun ; when they appear half full, they are in those parts whicliIre over against the sun ; when horned, in those parts which lie betweenthe earth and the sun ; and sometimes they pass over the sun s disk, whendirectly interposed between the eirth and the sun.And Venus, with a motion almost uniform, describes an orb nearly circular and concentric with the sun.But Mercury, with a more eccentric motion, makes remarkable approaches to the sun, and goes off again by turns ; but it is always swifteras it is near to the sun, and therefore by a radius drawn to the sun stilldescribes areas proportional to the times.Lastly, that the earth describes about the sun, or the sun about theearth, by a radius from the one to the other, areas exactly proportional tothe times, is demonstrable from the apparent diameter of the sun compared with its apparent motion.These are astronomical experiments ; from which it follows, by Prop. I,516 THE SYSTEM OF THE WORLD.11, III, in the first Book of our Pn/triples, and their Corollaries (p.213, 214). that there are centripetal forces actually directed (either accurately or without considerable error) to the centres of the earth, of Jupiter, of S.iturn, and of the sun. In Mercury, Venus, Mars, and the lesserplanets, wheie experiments are wanting, the arguments from analogy mustbe allowed in their place.That those forces (p. 212, 213, 214) decrease in the duplicate proportion of the distances from the centre of every planet, appears by Cor. VI,Prop. IV, Book 1;for the periodic times of the satellites of Jupiter areone to another (p. 386, 387) in the sesquiplicate proportion of their distances from the centre of this planet.This proportion has been long ago observed in those satellites ; and Mr.Flamsted, who had often measured their distances from Jupiter by themicrometer, and by the eclipses of the satellites, wrote to me, that it holdsto all the accuracy that possibly can be discerned by our senses. And hesent me the dimensions of their orbits taken by the micrometer, a*nd reduced to the mean distance of Jupiter from the earth, or from the sun,together with the times of their revolutions, as follows :Wherce the sesquiplicate proportion may be easily seen. For example ;the 16(f 18h. 05 13" is to the time l d. 18h. 28 36" as 493i" x V 493i"to 108 X V 108", neglecting those small fractions which, in observing,cannot ./e certainly determined.Befo e the invention of the micrometer, the same distances vrere determined 7 。 semi-diameters of Jupiter thus :After the invention of the micrometer :THE SYSTEM OF THE WORLD. 6l7And the periodic times of those satellites, by the observations of Mr.Flamsted, are l d. 18h. 28 36"|3(l. 13". 17 54"| 7(1. 3h. 59 36"|16".IS11. 5 13". as above.And the distances thence computed are 5,578 | 8,878 | 14,168 | 24,968,accurately agreeing with the distances by observation.Cassini assures us (p. 388, 389) that the same proportion is observedin the circum-saturnal planets. But a longer course of observations isrequired before we can have a certain and accurate theory of those planets.In the circum-solar planets, Mercury and Venus, the same proportionholds with great accuracy, according to the dimensions of their orbs, asdetermined by the observations of the best astronomers.That Mars is revolved about the sun is demonstrated from the phaseswhich it shews, and the proportion of its apparent diameters (p. 388, 389,and 390) ; for from its appearing fall near conjunction with the sun, andgibbous in its quadratures, it is certain that it surrounds the sun.And since its diameter appears about five times greater when in opposition to the sun than when in conjunction therewith, and its distance fromthe earth is reciprocally as its apparent diameter, that distance will beabout five times less when in opposition to than when in conjunction withthe sun; but in both cases its distance from the sun will be nearly aboutthe same with the distance which is inferred from its gibbous appearancein the quadratures. And as it encompasses the sun at almost equal dist nces,but in respect of the earth is very unequally distant, so by radii drawnto the sun it describes areas nearly uniform ; but by radii drawn to theearth, it is sometimes swift, sometimes stationary, and sometimes retrograde.That Jupiter, in a higher orb than Mars, is likewise revolved about thesun, with a motion nearly equable, as well in distance as in the areas described, 1 infer thus.Mr. Flamsted assured me, by letters, that all the eclipses of the innermost satellite which hitherto have been well observed do agree with histheory so nearly, as never to differ therefrom by two minutes of time ;that in the outmost the error is little greater ;in the outmost but one,scarcely three times greater ;that in the innermost but one the differenceis indeed much greater, yet so as to agree as nearly with his computation?as the moon does with the common tables ; and that he computes thoseeclipses only from the mean motions corrected by the equation of light discovered and introduced by Mr. Rower. Supposing, then, that the theorydiffers by a less error than that of 2 from the motion of the outmost satellite as hitherto described, and taking as the periodic time 16 1. 18h. 5 13"to 2 in time, so is the whole circle or 360 to the arc 148", the error olMr. Flamsted s computation, reduced to the satellite s orbit, will be lessthan 1 48";that is, the longitude of the satellite, as seen from tlie centreof Jupiter; will be determined with a less error than 1 48". But when518 THE SYSTEM OF THE WORLD.the satellite is in the middle of the shadow, that longitude is the same withthe heliocentric longitude of Jupiter ; and, therefore, the hypothesis whichMr. Flamsted follows, viz., the Copernican, as improved by Kepler, andfas to the motion of Jupiter) lately corrected by himself, rightly representsthat longitude within a less error than 1 48"; but by this longitude, together with the geocentric longitude, which is always easily found, the distance of Jupiter from the sun is determined ; which must, therefore, be thevery same with that which the hypothesis exhibits. For that greatest errorof I 48" that can happen in the heliocentric longitude is almost insensible, and quite to be neglected, and perhaps may arise from some yet undiscovered eccentricity of the satellite : but since both longitude and distanceare rightly determined, it follows of necessity that Jupiter, by radii drawnto the sun. describes areas so conditioned as the hypothesis requires, that is.proportional to the times.And the same thing may be concluded of Saturn from his satellite, bythe observations of Mr. Huygens and Dr. Halley ; though a longer seriesof observations is yet wanting to confirm the thing, and to bring it undera sufficiently exact computation.For if Jupiter was viewed from the sun, it would never appear retrograde nor stationary, as it is seen sometimes from the earth, but always togo forward with a motion nearly uniform (p. 389). And from the verygreat inequality of its apparent geocentric motion, we infer (by Prop. IllCor. IV) that the force by which Jupiter is turned out of a rectilinear course,and made to revolve in an orb, is not directed to the centre of the earth.And the same argument holds good in Mars and in Saturn. Another centreof these forces is therefore to be looked for (by Prop. II and III, and theCorollaries of the latter), about which the areas described by radii intervening may be equable ; and that this is the sun, we have proved alreadyin Mars and Saturn nearly, but accurately enough in Jupiter. It may bealledged that the sun and planets are impelled by some other force equallyand in the direction of parallel lines ; but by such a force (by Cor. VI ofthe Laws of Motion) no change would happen in the situation of theplanets one to another, nor any sensible eifect follow : but our business iswith the causes of sensible effects. Let us, therefore, neglect every suchforce as imaginary and precarious, and of no use in the phenomena of theheavens ; and the whole remaining force by which Jupiter is impelled willbe directed (by Prop. Ill, Cor. I) to the centre of the sun.The distances of the planets from the sun come out the same, whether,with Tycho, we place the earth in the centre of the system, or the sun withCopernicus : and we have already proved that these distances are true ir.Jupiter.Kepler and Bullialdiis have, with great care (p. 388), determined thelistances of the planets from the sun ; and hence it is that their table.-?THE SYSTEM OF THE WORLD. 519agree best with the heavens. And in all the planets, in Jupiter and Mars,in Saturn and the earth, as well as in Venus and Mercury, the cubes of theirdistances are as the squares of their periodic times ; and therefore (by Cor.VI, Prop. IV) the centripetal circum-solar force throughout all the planetary regions decreases in the duplicate proportion of the distances from thesun. In examining this proportion, we are to use the mean distances, orthe transverse semi-axes of the orbits (by Prop. XV). arid to neglect thoselittle fractions, which, in denning the orbits, may have arisen from the insensible errors of observation, or may be ascribed to other causes which weshall afterwards explain. And thus we shall always find the said proportion to hold exactly; for the distances of Saturn, Jupiter, Mars, the Earth,Venus, and Mercury, from the sun, drawn from the observations of astronomers, are, according to the computation of Kepler, as the numbers95 LOGO, 519650, 152350, 100000, 72400, 3S806; by the computation of/iHllialdus, as the numbers 95419S, 522520, 152350, 100000, 72393,38585 ; and from the periodic times they come out 953806, 520116, 152399,100000, 72333, 38710. Their distances, according to Kepler andKtillwldus, scarcely differ by any sensible quantity, and where theydiffer most the distances drawn from the periodic times, fall in between them.That the circum-terrestrial force likewise decreases in the duplicate proportion of the distances, I infer thus.The mean distance of the moon from the centre of the earth, is, in semidiametersof the earth, according to Ptolemy, Kepler in his Ephemerides,Bidliuldus, Hevelius, and Ricciolns, 59 ; according to Flamsted, 59| ;according to Tycho, 56 1;to Vendelin, 60 ; to Copernicus, 60 1 : to Kircher,62i (p. 391, 392, 393).Cut Tycho, and all that follow his tables of refraction, making therefractions of the sun and moon (altogether against the nature of light)to exceed those of the fixed stars, and that by about four or five minutesin the horizon, did thereby augment the horizontal parallax of the moonby about the like number of minutes ; that is, by about the 12th or 15thpart of the whole parallax. Correct this error, and the distance will become 60 or 61 semi-diameters of the earth, nearly agreeing with whatothers have determined.Let us, then, assume the mean distance of the moon 60 semi-diametersof the earth, and its periodic time in respect of the fixed stars 27d. 7h. 43 ,as astronomers have determined it. And (by Cor. VI, Prop. IV) a bodyrevolved in our air, near the surface of the earth supposed at rest, bymeans of a centripetal force which should be to the same force at the distance of the moon in the reciprocal duplicate proportion of the distancesfrom the centre of the earth, that is, as 3600 to 1, would (secluding theresistance of the air) complete a revolution in lh. 24 27".Suppose the circumference of the earth to be 123249600 Paris feet; ar52C THE SYSTEM OF THE WORLD.has been determined by the late mensuration of the French (vide p. 406) ;then the sume body, deprived of its circular motion, and falling by theimpulse of the same centripetal force as before, would, in one second oftime, describe 15-^ Paris feet.This we infer by a calculus formed upon Prop. XXXYI, and it agreeswith what we observe in all bodies about the earth. For by the experiments of pendulums, and a computation raised thereon, Mr. Hnygens hasdemonstrated that bodies falling by all that centripetal force with which(of whatever nature it is) they are impelled near the surface of the earth,do, in one second of time, describe 15 T^ Paris feet.But if the earth is supposed to move, the earth and moon together (byCor. IV of the Laws of Motion, and Prop. LVID will be revolved abouttheir common centre of gravity. Ana the moon (by Prop. LX) will inthe same periodic time, 27 1. 7h. 43 , with the same circum terrestrial forcediminished in the duplicate proportion of the distance, describe an orbitwhose semi-diameter is to the semi-diameter of the former orbit, that is, to60 semi-diameters of the earth, as the sum of both the bodies of the earthand moon to the first of two mean proportionals between this sum and thebody of the earth;that is, if we suppose the moon (on account of itsmean apparent diameter 31^ ) to be about ^ of the earth, as 43 to^ 42 a-43|2, or as about 128 to 127. And therefore the semi-diameterof the orbit, that is, the distance between the centres of the moon andearth, will in this case be 60^ semi-diameters of the earth, almost the samewith that assigned by Copernicus, which the Tychonic observations by nomeans disprove ; and, therefore, the duplicate proportion of the decrementof the force holds good in this distance. I have neglected the incrementof the orbit which arises from the action of the sun as inconsiderable ;but if that is subducted, the true distance will remain about 60|- semidiametersof the earth.But farther (p. 390) ; this proportion of the decrement of the forces isconfirmed from the eccentricity of the planets, and the very slow motionof their apses ;for (by the Corollaries of Prop. XLV) in no other proportioncould the circum-solar planets once in every revolution descend totheir least and once ascend to their greatest distance from the sun, and theplaces of those distances remain immoveable. A small error from the duplicate proportion would produce a motion of the apses considerable inevery revolution, but in many enormous.But now, after innumerable revolutions, hardly any such motion ha&been perceived in the orbs of the circum-solar planets. Some astronomersaffirm that there is no such motion; others reckon it no greater than whatmay easily arise from the causes hereafter to be assigned, and is of no moment in the present question.THE SYSTEM 01 THE WORLD. 521We may even neglect the motion of the moon s apsis (p. 390, 391), whichis far greater than in the circum-solar planets, amounting in every revolution to three degrees ; and from this motion it is demonstrable that thecircum-terrestrial force decreases in no less than the duplicate, but far lessthan the triplicate proportion of the distance ; for if the duplicate proportion was gradually changed into the triplicate, the motion of the apsiswould thereby increase to infinity; and, therefore, by a very small mutation, would exceed the motion of the moon s apsis. This slow motion arisesfrom the action of the circum-solar force, as we shall afterwards explain.But, secluding this cause, the apsis or apogeon of the moon will be fixed,and the duplicate proportion of the decrease of the circum-terrestrial forcein different distances from the earth will accurately take place.Now that this proportion has been established, we may compare theforces of the several planets among themselves (p. 391).In the mean distance of Jupiter from the earth, the greatest elongationof the outmost satellite from Jupiter s centre (by the observations of Mr.Flamsted] is 8 13"; and therefore the distance of the satellite from thecentre of Jupiter is to the mean distance of Jupiter from tne centre of thesun as 124 to 52012, but to the mean distance of Venus from the centreof the sun as 124 to 7234; and their periodic times are 16 d. and 224fd;and from hence (according to Cor. II, Prop. IV), dividing the distances bythe squares of the times, we infer that the force by which the satellite isimpelled towards Jupiter is to the force by which Venus is impelled towards the sun as 442 to 143 ; and if we diminish the force by which thesatellite is impelled in the duplicate proportion of the distance 124 to7234, we shall have the circum-jovial force in the distance of Venus fromthe sun to the circum-solar force by which Venus is impelled as yW to143, or as 1 to 1100; wherefore at equal distances the circum-solar forceis 1100 times greater than the circum-jovial.And, by the like computation, from the periodic time of the satellite otSaturn 15(l. 22h. and its greatest elongation from Saturn, while that planetis in its mean distance from us, 3 20",it follows that the distance of thissatellite from Saturn s centre is to the distance of Venus from the sun as92| to 7234; and from thence that the absolute circum-solar force is 2360times greater than the absolute circum-saturnal.From the regularity of the heliocentric and irregularity of the geocentric motions of Venus, of Jupiter, and the other planets, it is evident (byCor. IV, Prop. Ill) that the circum-terrestrial force, compared with the circum-solar, is very small.Ricciolus and Vendelin have severally tried to determine the sun s parallax from the moon s dichotomies observed by the telescope, and they agreethat it does not exceed half a minute.Kepler, from Ti/cho s observations and his own, found the parallax ofo22 THE SYSTEM OF THE WORLD.Mars insensible, even in opposition to the sun, when that parallax is something greater than the sun s.Flamsted attempted the same parallax with the micrometer in the perigeonposition of Mars, but never found it above 25"; and thence conclud