自然哲学的数学原理-43

COR. 3. All the planets do mutually gravitate towards one another, byCor. 1 and 2. And hence it is that Jupiter and Saturn, when near their394 THE MATHEMATICAL PRINCIPLES [BOOK IIIconjunction; by their mutual attractions sensibly disturb each other s ?n>tions. So the sun disturbs the motions of the moon ; and both sun inimoon disturb our sea, as we shall hereafter explain.SCHOLIUM.The force which retains the celestial bodi in their orbits has beenhitherto called centripetal force; but it being now made plain that it canbe 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 orbitwill extend itself to all the planets, by Rule I, II, and IV.PROPOSITION VI. THEOREM VI.That all bodies gravitate towards every planet ; and that the weights ofbodies towards any the same planet, at equal distances from the centreof the planet, are proportional to the quantities of matter which theyseverally contain.It has been, now of a long time, observed by others, that all sorts ofheavy bodies (allowance being made for the inequality of retardation whichthey suffer from a small power of resistance in the air) descend to theearth from equal heights in equal times; and that equality of times wemay distinguish to a great accuracy, by the help of pendulums. I tried thething in gold, silver, lead, glass, sand, eommpn salt, wood, water, and wheat.I provided two wooden boxes, round and equal : I filled the one with wood,and suspended an equal weight of gold (as exactly as I could) in the centreof oscillation of the other. The boxes hanging by equal threads of 11 feetmade a couple of pendulums perfectly equal in weight and figure, andequally receiving the resistance of the air. And, placing the one by theother, I observed them to play together forward and backward, for a longtime, wi h equal vibrations. And therefore the quantity of matte* : n thegold (by Cor. 1 and 6, Prop. XXIV, Book II) was to the quantity ot matter in the wood as the action of the motive force (or vis tnotrix) upon allthe gold to the action of the same upon all the wood ; that is, as the weightof the one to the weight of the other : and the like happened in the otherbodies. By these experiments, in bodies of the same weight, 1 could manifestly have discovered a difference of matter less than the thousandth partof the whol^, had any such been. But, without all doubt, the nature ofgravity towards the planets is the same as towards the earth. For, shouldwe imagine our terrestrial bodies removed to the orb of the moon, andthere, together with the moon, deprived of all motion, to be let go, so as tofall together towards the earth, it is certain, from what we have demonstrated before, that, in equal times, they would describe equal spaces with themoon, and of consequence are to the moon, in quantity of matter, as theirweights to its weight. Moreover, since the satellites of Jupiter performHOOK ill.] or NVTURAL PHILOSOPHY, 395their revolutions in times which observe the sesquiphiate pr portion oltheir distances from Jupiter s centre, their accelerative gravities towardsJupiter will be reciprocally as the squares of their distances from Jupiter scentre; that is, equal, at equal distances. And, therefore, these satellites,if supposed to fall towards Jupiter from equal heights, would describe equalspaces in equal times, in like manner as heavy bodies do on our earth.And, by the same argument, if the circumsolar planets were supposed to belet fall at equal distances from the sun, they would, in their descent towardsthe sun, describe equal spaces in equal times. But forces which equallyaccelerate unequal bodies must be as those bodies : that is to sa_y, the weights;f the planets towards the sun must be as their quantities of matter,further, that the weights of Jupiter and of his satellites towards the sunare proportional to the several quantities of their matter, appears from theexceedingly regular motions of the satellites (by Cor. 3, Prop. LXV, Book1). For if some of those bodies were more strongly attracted to the sun inproportion to their quantity of matter than others, the motions of the satellites would be disturbed by that inequality of attraction (by Cor.^, Prop.LXV, Book I). If, at equal distances from the sun, any satellite, in proportion to the quantity of its matter, did gravitate towards the sun with aforce greater than Jupiter in proportion to his, according to any given proportion, suppose of d to e ; then the distance between the centres of the sunand of the satellite s orbit would be always greater than the distance between the centres of the sun and of Jupiter nearly in the subduplicate ofthat proportion : as by some computations I have found. And if the satellite did gravitate towards the sun with a force, lesser in the proportion of eto d, the distance of the centre of the satellite s orb from the sun would beless than the distance of the centre of Jupiter from the sun in the subduplicate of the same proportion. Therefore if, at equal distances from thesun, the accelerative gravity of any satellite towards the sun were greateror less than the accelerative gravity of Jupiter towards the sun but by one T oV 7part of the whole gravity, the distance of the centre of the satellite s orbitfrom the sun would be greater or less than the distance of Jupiter from thesun by one ^oVo Part of the whole distance; that is, by a nf h part of thedistance of the utmost satellite from the centre of Jupiter ; an eccentricityof the orbit which would be very sensible. But the orbits of the satellitesare concentric to Jupiter, and therefore the accelerative gravities of Jupiter,and of all its satellites towards the sun, are equal among themselves. Andby the same argument, the weights of Saturn and of his satellites towardsthe sun, at equal distances from the sun, are as their several quantities ofmatter ; and the weights of the moon and of the earth towards the sun areeither none, or accurately proportional to the masses of matter which theycontain. But some they are, by Cor. 1 and 3, Prop. V.But further ; the weights of all the parts of every planet f awards any other396 THE MATHEMATICAL PRINCIPLES [BOOK II]planet are one to another as the matter in the several parts; for if someparts did gravitate more, others less, than for the quantity of their matter,then the whole planet, according to the sort of parts with which it mostabounds, would gravitate more or less than in proportion to the quantity ofmatter in the whole. Nor is it of any moment whether these parts areexternal or internal;for if, for example, we should imagine the terrestrialbodies with us to be raised up to the orb of the moon, to be there comparedwith its body : if the weights of such bodies were to the weights of the external parts of the moon as the quantities of matter in the one and in theother respectively but to the weights of the internal parts in a greater orless proportion, then likewise the weights of those bodies would be to theweight of the whole moon in a greater or less proportion; against whatwe have shewed above.COR. 1. Hence the weights of bodies do not depend upon their formsand textures ; for if the weights could be altered with the forms, theywould be greater or less, according to the variety of forms, in equal matter ;altogether against experience.COR. 2. Universally, all bodies about the earth gravitate towards theearth ; and the weights of all, at equal distances from the earth s centre.are as the quantities of matter which they severally contain. This is thequality of all bodies within the reach of our experiments ; and therefore(by Rule III) to be affirmed of all bodies whatsoever. If the ather, or anjother body, were either altogether void of gravity, or were to gravitate lesrin proportion to its quantity of matter, then, because (according to Aristotle, Des Carles, and others) there is no difference betwixt that and otherbodies but in mere form of matter, by a successive change from form toform, it might be changed at last into a body of the same condition withthose which gravitate most in proportion to their quantity of matter ; and,on the other hand, the heaviest bodies, acquiring the first form of thatbody, might by degrees quite lose their gravity. And therefore the weightswould depend upon the forms of bodies, and with those forms might bechanged : contrary to what was proved in the preceding Corollary.COR. 3. All spaces are not equally full; for if all spaces were equallyfull, then the specific gravity of the fluid which fills the region of the air,on account of the extreme density of the matter, would fall nothing shortof the specific gravity of quicksilver, or gold, or any other the most densebody ; and, therefore, neither gold, nor any other body, could descend inair;for bodies do not descend in fluids, unless they are specifically heavierthan the fluids. And if the quantity of matter in a given space can, byany rarefaction, be diminished, what should hinder a diminution toinfinity ?COR. 4. If all the solid particles of all bodies are of the same density,nor can be rarefied without pores, a void, space, or -acuum must be grantedBOOK Ill.J OF NATURAL PHILOSOPHY. 397By bodies of the same density, I mean those whose vires inertia are in theproportion of their bulks.COR. 5. The power of gravity is of a different nature from the power ofmagnetism ;for the magnetic attraction is not as the matter attracted.Some bodies are attracted more by the magnet ;others less; most bodiesnot at all. The power of magnetism in one and the same body may beincreased and diminished ; and is sometimes far stronger, for the quantityof matter, than the power of gravity ; and in receding from the magnetdecreases not in the duplicate but almost in the triplicate proportion of thedistance, as nearly as I could judge from some rude observations.PROPOSITION VII. THEOREM VII.That there is a power of gravity tending to all bodies, proportional tothe several quantities of matter which they contain.That all the planets mutually gravitate one towards another, we haveproved before ;as well as that the force of gravity towards every one of them,considered apart, is reciprocally as the square of the distance of places fromthe centre of the planet. And thence (by Prop. LXIX, Book I, and itsCorollaries) it follows, that the gravity tending towards all the planets isproportional to the matter which they contain.Moreover, since all the parts of any planet A gravitate towards anyother planet B ; and the gravity of every part is to the gravity of thewhole as the matter of the part to the matter of the whole ; and (by LawIII) to every action corresponds an equal re-action ; therefore the planet Bwill, on the other hand, gravitate towards all the parts of the planet A ;and its gravity towards any one part will be to the gravity towards thewhole as the matter of the part to the matter of the whole. Q.E.D.COR, 1. Therefore the force of gravity towards any whole planet arisesfrom, and is compounded of, the forces of gravity towards all its parts.Magnetic and electric attractions afford us examples of this;for all attraction towards the whole arises from the attractions towards the severalparts. The thing may be easily understood in gravity, if we consider agreater planet, as formed of a number of lesser planets, meeting together inone globe ; for hence it would appear that the force of the whole mustarise from the forces of the component parts. If it is objected, that, according to this law, all bodies with us must mutually gravitate one towards another, whereas no such gravitation any where appears, I answer,that since the gravitation towards these bodies is to the gravitation towardsthe whole earth as these bodies are to the whole earth, the gravitation towards them must be far less than to fall under the observation of our senses.COR. 2. The force of gravity towards the several equal particles of anybody is reciprocally as the square of the distance of places from the particles ; as appears from Cor. 3, Prop. LXXIV, Book I.39S THE MATHEMATICAL PRINCIPLES [BOOK IIIPROPOSITION VIII. THEOREM VIII.Tn two spheres mutually gravitating each towards the other, if tlie matterin places on all sides round about and equi-distantfrom the centres issimilar, the weight of either sphere towards the other will be reciprocally as the square of the distance between their centres.After I had found that the force of gravity towards a whole planet didarise from and was compounded of the forces of gravity towards all itsparts, and towards every one part was in the reciprocal proportion of thesquares of the distances from the part, I was yet in doubt whether that reciprocal duplicate proportion did accurately hold, or but nearly so, in thetotal force compounded of so many partial ones; for it might be that theproportion which accurately enough took place in greater distances shouldbe wide of the truth near the surface of the planet, where the distances ofthe particles are unequal, and their situation dissimilar. But by the helpof Prop. LXXV and LXXVI, Book I, and their Corollaries, I was at lastsatisfied of the truth of the Proposition, as it now lies before us.COR. 1. Hence we may find and compare together the weights of bodiestowards different planets ;for the weights of bodies revolving in circlesabout planets are (by Cor. 2, Prop. IV, Book I) as the diameters of thecircles directly, and the squares of their periodic times reciprocally ; andtheir weights at the surfaces of the planets, or at any other distances fromtheir centres, are (by this Prop.) greater or less in the reciprocal duplicateproportion of the distances. Thus from the periodic times of Venus, revolving about the sun, in 224<J. 16fh, of the utmost circumjovial satelliterevolving about Jupiter, in 16 . 10 -?/. ; of the Huygenian satellite aboutSaturn in 15d. 22fh.; and of the moon about the earth in 27d. 7h. 43 ;compared with the mean distance of Venus from the sun, and with thegreatest heliocentric elongations of the outmost circumjovial satellitefrom Jupiter s centre, 8 16"; of the Huygenian satellite from the centreof Saturn, 3 4";arid of the moon from the earth, 10 33" : by computation I found that the weight of equal bodies, at equal distances from thecentres of the sun, of Jupiter, of Saturn, and of the earth, towards the sun,Jupiter, Saturn, and the earth, were one to another, as 1, T ^VT> ^oVr? an^___i___respectively. Then because as the distances are increased or diminished, the weights are diminished or increased in a duplicate ratio, theweights of equal bodies towards the sun, Jupiter, Saturn, and the earth,at the distances 10000, 997, 791, and 109 from their centres, that is, at theirvery superficies, will be as 10000, 943, 529, and 435 respectively. Howmuch the weights of bodies are at the superficies of the moon, will beshewn hereafter.COR. 2. Hence likewise we discover the quantity of matter in the several.BOOK II1.J OF NATURAL PHILOSOPHY. 39(.*planets; for their quantities of matter are as the forces of gravity at equaidistances from their centres; that is, in the sun, Jupiter, Saturn, and theearth, as 1, TO FTJ a-oVr? anc^ TeVaja respectively. If the parallax of thesun be taken greater or less than 10" 30 ", the quantity of matter inthe earth must be augmented or diminished in the triplicate of that proportion.COR. 3. Hence also we find the densities of the planets ;for (by Prop.LXXII, Book I) the weights of equal and similar bodies towards similarspheres are, at the surfaces of those spheres, as the diameters of the spheres 5and therefore the densities of dissimilar spheres are as those weights appliedto the diameters of the spheres. But the true diameters of the Sun, .Jupiter, Saturn, and the earth, were one to another as 10000, 997, 791, arid109; and the weights towards the same as 10000, 943, 529, and 435 respectively ; and therefore their densities are as 100. 94|, 67, and 400. Thedensity of the earth, which comes out by this computation, does not dependupon the parallax of the sun, but is determined by the parallax of themoon, and therefore is here truly defined. The sun, therefore, is a littledenser than Jupiter, and Jupiter than Saturn, and the earth four timesdenser than the sun ; for the sun, by its great heat, is kept in a sort ofa rarefied state. The moon is denser than the earth, as shall appear afterward.COR. 4. The smaller the planets are, they are, cccteris parilms, of somuch the greater density ;for so the powers of gravity on their severalsurfaces come nearer to equality. They are likewise, cccteris paribiis, ofthe greater density, as they are nearer to the sun. So Jupiter is moredense than Saturn, and the earth than Jupiter ;for the planets were to beplaced at different distances from the sun, that, according to their degreesof density, they might enjoy a greater or less proportion to the sun s heat.Our water, if it were removed as far as the orb of Saturn, would be converted into ice, and in the orb of Mercury would quickly fly away in vapour ;for the light of the sun, to which its heat is proportional, is seventimes denser in the orb of Mercury than with us : and by the thermometerI have found that a sevenfold heat of our summer sun will make waterboil. Nor are we to doubt that the matter of Mercury is adapted to itsheat, and is therefore more dense than the matter of our earth ; since, in adenser matter, the operations of Nature require a stronger heat.PROPOSITION IX. THEOREM IX.That the force of gravity, considered downward from t/ie surfaceof the planets decreases nearly in the proportion of the distances fromtheir centres.If the matter of the planet were of an uniform density, this Proposition would be accurately true (by Prop. LXXIII. Book I). The error,100 THE MATHEMATICAL PRINCIPLES [BOOK IIItherefore, can be no greater than what may arise from the inequality ofthe density.PROPOSITION X. THEOREM X.That the motions of the planets in the heavens may subsist an exceedinglylong time.In the Scholium of Prop. XL, Book II, I have shewed that a globe ofwater frozen into ice, and moving freely in our air, in the time that it woulddescribe the length of its semi-diameter, would lose by the resistance of theair 3。6 part of its motion; and the same proportion holds nearly in allglobes, how great soever, and moved with whatever velocity. But that ourglobe of earth is of greater density than it would be if the wholeconsisted of water only, I thus make out. If the whole consisted ofwater only, whatever was of less density than water, because of its Ivssspecific gravity, would emerge and float above. And upon this account, ifa globe of terrestrial matter, covered on all sides with water, was less densethan water, it would emerge somewhere ; and, the subsiding water fallingback, would be gathered to the opposite side. And such is the conditionof our earth, which in a great measure is covered with seas. The earth, ifit was not for its greater density, would emerge from the seas, and, according to its degree of levity, would be raised more or less above their surface,the water of the seas flowing backward to the opposite side. By the sameargument, the spots of the sun, which float upon the lucid matter thereof.are lighter than that matter ; and, however the planets have been formedwhile they were yet in fluid masses, all the heavier matter subsided to thecentre. Since, therefore, the common matter of our earth on the surfacethereof is about twice as heavy as water, and a little lower, in mines, isfound about three, or four, or even five times more heavy, it is probable thatthe quantity of the whole matter of the earth may be five or six timesgreater than if it consisted all of water ; especially since I have beforeshewed that the earth is about four times more dense than Jupiter. If,therefore, Jupiter is a little more dense than water, in the space of thirtydays, in which that planet describes the length of 459 of its semi-diameters, it would, in a medium of the same density Avith our air, lose almost atenth part of its motion. But since the resistance of mediums decreasesin proportion to their weight or density, so that water, which is 13| timeslighter than quicksilver, resists less in that proportion ; and air, which is860 times lighter than water, resists less in the same proportion ; thereforein the heavens, where the weight of the medium in which the planets moveis immensely diminished, the resistance will almost vanish.It is shewn in the Scholium of Prop. XXII, Book II, that at the heightof 200 miles above the earth the air is more rare than it is at the superficies of the earth in the ratio of 30 to 0,0000000000003999, or asBOOK III.] OF NATURAL PHILOSOPHY. 40175000000000000 to 1 nearly. And hence the planet Jupiter, revolving ina medium of the same density with that superior air, would not lose by theresistance of the medium the 1000000th part of its motion in 1000000years. In the spaces near the earth the resistance is produced only by theair, exhalations, and vapours. When these are carefully exhausted by theair-pump from under the receiver, heavy bodies fall within the receiver withperfect freedom, and without the t sensible resistance: gold itself, andthe lightest down, let fall together, will descend with equal velocity; andthough they fall through a space of four, six, and eight feet, they will cometo the bottom at the same time; as appears from experiments. And therefore the celestial regions being perfectly void of air and exhalations, theplanets and comets meeting no sensible resistance in those spaces will continue their motions through them for an immense tract of time.HYPOTHESIS I.That the centre of the system of the world is immovable.This is acknowledged by all, while some contend that the earth,others that the sun, is fixed in that centre. Let us see what may fromhence follow.PROPOSITION XL THEOREM XI.That the common, centre of gravity of the earth, the sun, and all theplanets, is immovable.For (by Cor. 4 of the Laws) that centre either is at rest, or moves uniformly forward in a right line; but if that centre moved, the centre of theworld would move also, against the Hypothesis.PROPOSITION XII. THEOREM XII.That the sun is agitated by a perpetual motion, but never recedes jarfrom the common, centre of gravity of all the planets.For since (by Cor. 2, Prop. VIII) the quantity of matter in the sun is tothe quantity of matter in Jupiter as 1067 to 1; and the distance of Jupiter from the sun is to the semi-diameter of the sun in a proportion but asmall matter greater, the common centre of gravity of Jupiter and the sunwill fall upon a point a little without the surface of the sun. By the sameargument, since the quantity of matter in the sun is to the quantity ofmatter in Saturn as 3021 to 1, and the distance of Saturn from the sun isto the semi-diameter of the sun in a proportion but a small matter less,the common centre of gravity of Saturn and the sun will fall upon a pointa little within the surface of the sun. And, pursuing the principles of thiscomputation, we should find that though the earth and all the planets wereplaced on one side of the sun, the distance of the common centre of gravityof all from the centre of the sun would scarcely amount to one diameter of26102 THE MATHEMATICAL PRINCIPLES [BOOK IIIthe sun. In other cases, the distances of those centres are always less : andtherefore, since that centre of gravity is in perpetual rest, the sun, according to the various positions of the planets, must perpetually be moved everyway, but will never recede far from that centre.Con. Hence the common centre of gravity of the earth, the sun, and allthe planets, is to be esteemed the centre of the world ; for since the earth,the sun, and all the planets, mutually gravitate one towards another, andare therefore, according to their powers of gravity, in perpetual agitation,as the Laws of Motion require, it is plain that their moveable centres cannot be taken for the immovable centre of the world. If that body were tobe placed in the centre, towards which other bodies gravitate most (according to common opinion), that privilege ought to be allowed to the sun; butsince the sun itself is moved, a fixed point is to be chosen from which thecentre of the sun recedes least, and from which it would recede yetless if the body of the sun were denser and greater, and therefore less apt