自然哲学的数学原理-57

ed the sun s parallax at most 10".Whence it follows that the distance of the moon from the earth bearsno greater proportion to the distance of the earth from the sun than 29 toIOUOO : nor to the distance of Venus from the sun than 29 to 7233.From which distances, together with the periodic times, by the methodabove explained, it is easy to infer that the absolute circum-soiar force isgreater than the absolute circum-terrestrial force at least 229400 times.And though we were only certain, from the observations of Ricciolusand Vcitdelin, that the sun s parallax was less than half a minute, yet fromthis it will follow that the absolute circum-solar force exceeds the absolutecircum-terrestrial force S500 times.By the like computations I happened to discover an analogy, that is observed between the forces and the bodies of the planets ; but, before I explain this analogy, the apparent diameters of the planets in their meandistances from the earth must be first determined.Mr. Flamsted (p. 387), by the micrometer, measured the diameter ofJupiter 40" or 41"; the diameter of Saturn s ring 50"; and the diameterof the sun about 32 13" (p. 387).But the diameter of Saturn is to the diameter of the ring, according toMr. Huygens and Dr. Halley, as 4 to 9; according to Gullet-ins, as 4 to10; and according to Hooke (by a telescope of 60 feet), as 5 to 12. Andfrom the mean proportion, 5 to 12, the diameter of Saturn s body is inferred about 21".Such as we have said are the apparent magnitudes; but. because of theunequal refrano-ibility of light, all lucid points are dilated by the telescope, and in the focus of the object-glass possess a circular space whosebreadth is about the 50th part of the aperture of the glass.It is true, that towards the circumference the light is Su rare as hardlyto move the sense;but towards the middle, where it is of greater density,and is sensible enough, it makes a small lucid circle, whose breadth variesaccording to the splendor of the lucid point, but is generally about the 3d,or 4th, or 5th part of the breadth of the whole.Let ABD represent the circle of the whole light; PQ the small circleof the denser and clearer light; C the centre of both; CA, CB, semi-diameters of the greater circle containing a right angle at C ; ACBE thesquare comprehended under these semi-diameters ; AB the diagonal of thatsquare; EGH an hyperbola with the centre C and asymptotes CA, CBPG a perpendicular erected from any point P of the line BC, and meetingthe hyperbola in G, and the right lines AB, AE, in K and F : and thTHE SYSTEM OF THE WORLD. 523density of the light in any place P, will, by my computation, be as theline FG, and therefore at the centre infinite, but near the circumferencevery small. And the whole light within the small circle PQ, is to thewithout as the area of the quadrilateral figure CAKP to the trianglePKB. And we are to understand the small circle PQ, to te thereterminated, where FG, the density of the light, begins to be less than whatis required to move the sense.Hence it was, that, at the distance of 191 382 feet, a fire of 3 feet in diameter, through a telescope of 3 feet, appeared to Mr. Picart of S" inbreadth, when it should have appeared only of 3" 14 "; and hence it isthat the brighter fixed stars appear through the telescope as of 5" or 6" indiameter, and that with a good full light ; but with a fainter light theyappear to run out to a greater breadth. Hence, likewise, it was that Hevelius,by diminishing the aperture of the telescope, did cut off a great partof the light towards the circumference, and brought the disk of the star tobe more distinctly defined, which, though hereby diminished, did yet appear as of 5" or 6" in diameter. But Mr. Hvygetis, only by clouding theeye-glass with a little smoke, did so effectually extinguish this scatteredlight, that the fixed stars appeared as mere points, void of all sensiblebreadth. Hence also it was that Mr. Huygens, from the breadth of bodiesinterposed to intercept the whole light of the planets, reckoned their diameters greater than others hav measured them by the micrometer : for the524 THE SYSTEM OF THE WORLD.scattered light, which could not be seen before for the stronger light of theplanet, when the planet is hid, appears every way farther spread. Lastly,from hence it is that the planets appear so small in the disk of the sun,being lessened by the dilated light. For to Hevelius, Galletius, and Dr.Halley, Mercury did not seem to exceed 12" or 15"; and Venus appearedto Mr. Crabtrie only 1 3 ;/;to Horrox but 1 12"; though by the mensurations of Hevelius and Hu&enius without the sun s disk, it ought tohave been seen at least 1 24". Thus the apparent diameter of the moon,which in 1 684, a few days both before and after the sun s eclipse, wasmeasured at the observatory of Paris 31 30", in the eclipse itself did notseem to exceed 30 or 30 05"; and therefore the diameters of the planetsare to be diminished when without the sun, and to be augmented whenwithin it, by some seconds. But the errors seem to be less than usual inthe mensurations that are made by the micrometer. So from the diameterof the shadow, determined by the eclipses of the satellites, Mr. Flamstedfound that the semi- diameter of Jupiter was to the greatest elongation ofthe outmost satellite as 1 to 24,903. Wherefore since that elongation is8 13 , the diameter of Jupiter will be 39^-" ; and, rejecting the scatteredlight, the diameter found by the micrometer 40" or 41" will be reduced to39|-" 5 and the diameter of Saturn 21" is to be diminished by the like correction, and to be reckoned 20", or something less. But (if I am not mistaken) the diameter of the sun, because of its stronger light, is to be diminished something more, and to be reckoned about 32 , or 32 6 1.That bodies so different in magnitude should come so near to an analogywith their forces, is not without some mystery (p. 400).It may be that the remoter planets, for want of heat, have not those metallic substances and ponderous minerals with which our earth abounds ;and that the bodies of Venus and Mercury, as they are more exposed to thesun s heat, are also harder baked, and more compact.For, from the experiment of the burning-glass, we see that the heat increases with the density of light ; and this density increases in the reciprocal duplicate proportion of the distance from the sun ; from whence thesan s heat in Mercury is proved to be sevenfold its heat in our summerseasons. But with this heat our water boils ; and those heavy fluids, quicksilver and the spirit of vitriol, gently evaporate, as I have tried by thethermometer ; and therefore there can be no fluids in Mercury but whatare heavy, and able to bear a great heat, and from which substances of greatdensity may be nourished.And why not, if God has placed different bodies at different distancesfrom the sun, so as the denser bodies always possess the nearer places, andeach body enjoys a degree of heat suitable to its condition, and proper forits nourishment? From this consideration it will best appear that theweights of all the planets are one to another as their forces.THE SYSTEM OF THE WORLD.But I should be glad the diameters of the planets were more accuratelymeasured ; and that may be done, if a lamp, set at a great distance, is madeto shine through a circular hole, and both the hole and the light of thelamp are so diminished that the spectrum may appear through the telescopejust like the planet, and may be defined by the same measure : then thediameter of the hole will be to its distance from the objective glass as thetrue diameter of the planet to its distance from us. The light of the lampmay be diminished by the interposition either of pieces of cloth, or ofsmoked glass.Of kin to the analogy we have been describing, there is another observedbetween the forces and the bodies attracted (p. 395, 396, 397). Since theaction of the centripetal force upon the planets decreases in the duplicateproportion of the distance, and the periodic time increases in the sesquiplicatethereof, it is evident that the actions of the centripetal force, andtherefore the periodic times, would be equal in equal planets at equal distances from the sun ; and in equal distances of unequal planets the totalactions of the centripetal force would be as the bodies of the planets ; forif the actions were not proportional to the bodies to be moved, they couldnot equally retract these bodies from the tangents of their orbs in equaltimes : nor could the motions of the satellites of Jupiter be so regular, if itwas not that the circum-solar force was equally exerted upon Jupiter andall its satellites in proportion of their several weights. And the same thingis to be said of Saturn in respect of its satellites, and of our earth in respect of the moon, as appears from Cor. II and III, Prop. LXV. Arid,therefore, at equal distances, the actions of the centripetal force are equalupon all the planets in proportion of their bodies, or of the quantities ofmatter in their several bodies; and for the same reason must be the sameupon all the particles of the same size of which the planet is composed ; forif the action was greater upon some sort of particles than upon others thanin proportion to their quantity of matter, it would be also greater or lessupon the whole planets not in proportion to the quantity only, but likewise of the sort of the matter more copiously found in one and moresparingly in another.In such bodies as are found on our earth of very different sorts, I examined this analogy with great accuracy (p. 343, 344).If the action of the circum-terrestrial force is proportional to the bodiesto be moved, it will (by the Second Law of Motion) move them with equalvelocity in equal times, and will make all bodies let fall to descend throughequal spaces in equal times, and all bodies hung by equal threads to vibratein equal times. If the action of the force was greater, the times would beless ; if that was less, these would be greater.But it has been long ago observed by others, that (allowance being madefor the small resistance of the air) all bodies descend through equal spaces526 THE SYSTEM OF THE WORLD.in equal times ; and, by the help of pendulums, that equality of tim-es maybe distinguished to great exactness.1 tried the thing in gold, silver, lead, glass, sand, common salt wood,water, and wheat. I provided two equal wooden boxes. I filled the onewith wood, and suspended an equal weight of gold (as exactly as I could)in the centre of oscillation of the other. The boxes, hung by equal threadsof 11 feet, made a couple of pendulums perfectly equal in weight and figure, and equally exposed to the resistance of the air : and, placing the oneby the other, I observed them to play together forwards and backwards fora long while, with equal vibrations. And therefore (by Cor. 1 and VI,Prop. XXIV. Book II) the quantity of matter in the gold was to the quantity of matter in the wood as the action of the motive force upon all thegold to the action of the same upon all the wood ; that is, as the weight ofthe one to the weight of the other.And by these experiments, in bodies of the same weight, could have discovered a difference of matter less than the thousandth part of the whole.Since the action of the centripetal force upon the bodies attracted is, atequal distances, proportional to the quantities of matter in those bodies,reason requires that it should be also proportional to the quantity of matter in the body attracting.For all action is mutual, and (p. 83, 93. by the Third Law of Motion)makes the bodies mutually to approach one to the other, and therefore mustbe the same in both bodies. It is true that we may consider one body asattracting, another as attracted; but this distinction is more mathematicalthan natural. The attraction is really common of either to other, andtherefore of the same kind in both.And hence it is that the attractive force is found in both. The sun attracts Jupiter and the other planets ; Jupiter attracts its satellites ; and,for the same reason, the satellites act as well one upon another as upon Jupiter, and all the planets mutually one upon another.And though the mutual actions of two planets may be distinguishedand considered as two, by which each attracts the other, yet, as those actions are intermediate, they do not make two but one operation betweentwo terms. Two bodies may be mutually attracted each to the other bythe contraction of a cord interposed. There is a double cause of action,to wit, the disposition of both bodies, as well as a double action in so faras the action is considered as upon two bodies ; but as betwixt two bodiesit is but one single one. It is not one action by which the sun attractsJupiter, and another by which Jupiter attracts the sun ; but it is one action by which the sun and Jupiter mutually endeavour to approach eachthe other. By the action with which the sun attracts Jupiter, Jupiter andthe sun endeavours to come nearer together (by the Third Law of Motion) ; and by the action with which Jupiter attracts the sun, likewise JuTHE SYSTEM OF THE WORLD. 527pitcr and the sun endeavor to come nearer together. But the sun is notattracted towards Jupiter by a twofold action, nor Jupiter by a twofoldaction towards the sun ; but it is one single intermediate action, by whichboth approach nearer together.Thus iron draws the load-stone (p. 93), as well as the load-stonedraws the iron : for all iron in the neighbourhood of the load-stone drawsother iron. But the action betwixt the load-stone and iron is single, andis considered as single by the philosophers. The action of iron upon theload-stone, is, indeed, the action of the load-stone betwixt itself and theiron, by which both endeavour to come nearer together : and so it manifestly appears ; for if you remove the load-stone, the whole force of theiron almost ceases.Tn this sense it is that we are to conceive one single action to be exerted betwixt two planets, arising from the conspiring natures of both :and this action standing in the same relation to both, if it is proportionalto the quantity of matter in the one, it will be also proportional to thequantity of matter in the other.Perhaps it may be objected, that, according to this philosophy (p. 39S),all bodies should mutually attract one another, contrary to the evidenceof experiments in terrestrial bodies ; but I answer, that the experiments interrestrial bodies come to no account ; for the attraction of homogeneousspheres near their surfaces are (by Prop. LXXII) as their diameters.Whence a sphere of one foot in diameter, and of a like nature to theearth, would attract a small body placed near its surface with a force20UOOOOO times less than the earth would do if placed near its surface;but so small a force could produce no sensible effect. If two such sphereswere distant but by 1 of an inch, they would not, even in spaces void of528 THE SYSTEM OF THE WORLD.resistance, come together by the force of their mutual attraction in lessthan a month s time j and less spheres will come together at a rate yetslower, viz.. in the proportion of their diameters. Nay, whole mountainswill not be sufficient to produce any sensible effect. A mountain of anhemispherical figure, three miles high, and six broad, will not, by its attraction, draw the pendulum two minutes out of the true perpendicular :and it is only in the great bodies of the planets that these forces are to beperceived, unless we may reason about smaller bodies in manner following.Let ABCD (p. 93) represent the globe of the earth cut by any planeAC into two parts ACB, and A CD. The part ACB bearing upon thepart ACD presses it with its whole weight; nor can the part ACD sustainthis pressure and continue unmoved, if it is not opposed by an equal contrary pressure. And therefore the parts equally press each other by theirweights, that is, equally attract each other, according to the third Law ofMotion ; and, if separated and let go, would fall towards each other withvelocities reciprocally as the bodies. All which we may try and see in theload-stone, whose attracted part does not propel the part attracting, but isonly stopped and sustained thereby.Suppose now that ACB represents some small body on the earth s surface : then, because the mutual attractions of this particle, and of the remaining part ACD of the earth towards each other, are equal, but theattraction of the particle towards the earth (or its weight) is as the matterof the particle (as we have proved by the experiment of the pendulums),the attraction of the earth towards the particle will likewise be as thematter of the particle ; and therefore the attractive forces of all terrestrial bodies will be as their several quantities of matter.The forces (p. 396), which are as the matter in terrestrial bodies of allforms, and therefore are not mutable with the forms, must be found in allsorts of bodies whatsoever, celestial as well as terrestrial, and be in allproportional to their quantities of matter, because among all there is nodifference of substance, but of modes and forms only. But in the celestial bodies the same thing is likewise proved thus. We have shewn thatthe action of the circum-solar force upon all the planets (reduced to equaldistances) is as the matter of the planets ; that the action of the circumjovialforce upon the satellites of Jupiter observes the same law ; and thesame thing is to be said of the attraction of all the planets towards everyplanet : but thence it follows (by Prop. LXIX) that their attractive forcesare as their several quantities of matter.As the parts of the earth mutually attract one another, so do those ofall the planets. If Jupiter and its satellites were brought together, andformed into one globe, without doubt they would continue mutually toattract one another as before. And, on the other hand, if the body ofJupiter was broke into more globes, to be sure, these would no less attract.THE SYSTEM OF THE WORLD. 5293ne another than they do the satellites now. From these attractions it isthat the bodies of the earth and all the planets effect a spherical figure, andtheir parts cohere, and are not dispersed through the aether. But we havebefore proved that these forces arise from the universal nature of matter(p. 398), and that, therefore, the force of any whole globe is made up ofthe several forces of all its parts. And from thence it follows (by Cor.III, Prop. LXXIV) that the force of every particle decreases in the duplicate proportion of the distance from that particle ; and (by Prop. LXXIIIand LXXV) that the force of an entire globe, reckoning from the surfaceoutwards, decreases in the duplicate, but, reckoning inwards, in the simple proportion of the distances from the centres, if the matter of the globebe uniform. And though the matter of the globe, reckoning from thecentre towards the surface, is not uniform (p. 398, 399), yet the decrease inthe duplicate proportion of the distance outwards would (by Prop. LXXVI)take place, provided that difformity is similar in places round about atequal distances from the centre. And two sucli globes will (by the sameProposition) attract one the other with a force decreasing in the duplicateproportion of the distance between, their centres.Wherefore the absolute force of every globe is as the quantity of matterwhich the globe contains ; but the motive force by which every globe isattracted towards another, and which, in terrestrial bodies, we commonlycall their weight, is as the content under the quantities of matter in bothglobes applied to the square of the distance between their centres (by Cor.IV, Prop. LXXVI), to which force the quantity of motion, by which eachglobe in a given time will be carried towards the other, is proportional.And the accelerative force, by which every globe according to its quantityof matter is attracted towards another, is as the quantity of matter in thatother globe applied to the square of the distance between the centres ofthe two (by Cor. II, Prop. LXXVI): to which force, the velocity by whichthe attracted globe will, in a given time, be carried towards the other isproportional. And from these principles well understood, it will be noweasy to determine the motions of the celestial bodies among themselves.From comparing the forces of the planets one with another, we haveabove seen that the circum-solar does more than a thousand times exceedall the rest ; but by the action of a force so greab it is unavoidable but thatall bodies within, nay, and far beyond, the bounds of the planetary systemmust descend directly to the sun, unless by other motions they are impelledtowards other parts : nor is our earth to be excluded from the number ofsuch bodies : for certainly the moon is a body of the same nature with theplanets, and subject to the same attractions with the other planets, seeingit is by the circum-terrestrial force that it is retained in its orbit. Butthat the earth and moon are equally attracted towards the sun, we haveabove proved ; we have likewise before proved that all bodies are subject to530 THE SYSTEM OF THE WORLD.the said common laws of attraction. Nay, supposing any of those bodiesto be deprived of its circular motion about the sun, by having its distancefrom the sun, we may find (by Prop. XXXVI) in what space of time itwould in its descent arrive at the sun ;to wit, in half that periodic time in.vhich the body might be revolved at one half of its former distance; or ina space of time that is to the periodic time of the planet as 1 to4</2; asthat Venus in its descent would arrive at the sun in the space of 40 days,Jupiter in the space of two years and one month, and the earth and moontogether in the space of 66 days and 19 hours. But, since no such thinghappens, it must needs be, that those bodies are moved towards other parts(p. 75), nor is every motion sufficient for this purpose. To hinder such adescent, a due proportion of velocity is required. And hence depends theforce of the argument drawn from the retardation of the motions of theplanets. Unless the circum-solar force decreased in the duplicate ratio oftheir increasing slowness, the excess thereof would force those bodies to descend to the sun ; for instance, if the motion (c&teris paribns) was retardedby one half, the planet would be retained in its orb by one fourth of theformer circum-solar force, and by the excess of the other three fourthswould descend to the sun. And therefore the planets (Saturn, Jupiter,Mars, Venus, and Mercury) are not really retarded in their perigees, norbecome really stationary, or regressive with slow motions. All these arebut apparent, and the absolute motions, by which the planets continue torevolve in their orbits, are always direct, and nearly equable. But thatsuch motions are performed about the sun, we have already proved ; andtherefore the sun, as the centre of the absolute motions, is quiescent. Forwe can by no means allow quiescence to the earth, lest the planets in theirperigees should indeed be truly retarded, and become truly stationary andregressive, and so for want of motion should descend to the sun. Butfarther ; since the planets (Venus, Mars, Jupiter, and the rest) by radi:drawn to the sun describe regular orbits, and areas (as WTC have shewn)nearly and to sense proportional to the times, it follows (by Prop. III. andCor. Ill, Prop. LXV) that the sun is moved with no notable force, unlessperhaps wT ith such as all the planets are equally moved with, according totheir several quantities of matter, in parallel lines, and so the whole system is transferred in right lines. Reject that translation of the wholesystem, and the sun will be almost quiescent in the centre thereof. If thegun was revolved about the earth, and carried the other planets round aboutitself, the earth ought to attract the sun with a great force, but the circum-solar planets with no force producing any sensible effect, which iscontrary to Cor. Ill, Prop. LXV. Add to this, that if hitherto the earth,because of the gravitation of its parts, has been placed by most authors inthe lowermost region of the universe ; now, for better reason, the sun possessed of a centripetal force exceeding our terrestrial gravitation a thousandTHE SYSTEM OF THE WJP.I.D. 53Ltimes and more, ought to be depressed into the lowermost place, and to beheld for the centre of the system. And thus the true disposition of thewhole system will be more fully and more exactly understood.Because the fixed stars are quiescent one in respect of another (p. 401,4U2), we may consider the sun, earth, and planets, as one system of bodiescarried hither and thither by various motions among themselves; and thecommon centre of gravity of all (by Cor. IT of the Laws of Motion) willeither be quiescent, or move uniformly forward in a right line : in whichcase the whole system will likewise move uniformly forward in right lines.But this is an hypothesis hardly to be admitted ; and, therefore, setting itarfide, that common centre will be quiescent: and from it the sun is neverfar removed. The common centre of gravity of the sun and Jupiter fallson the surface of the sun ;and though all the planets were placed towardsthe same parts from the sun with Jupiter the common centre of the sunand all of them would scarcely recede twice as far from the sun s centre ;and, therefore, though the sun, according to the various situation of theplanets, is variously agitated, and always wandering to and fro with a slowmotion of libration, yet it never recedes one entire diameter of its own bodyfrom the quiescent centre of the whole system. But from the weights ofthe sun and planets above determined, and the situation of all among themselves, their common centre of gravity may be found ; and, this being given,the sun s place to any supposed time may be obtained.About the sun thus librated the other planets are revolved in elliptic

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