自然哲学的数学原理-55

to the computation of Mr. Bradley, Savilian Professor of Astronomy atOxford) was in T 14 16 . The inclination of the orbit to the plane ofthe ecliptic 49 59 . Its perihelion was in 8 12 15 20". Its periheliondistance from the sim 998651 parts, of which the radius of the orbis mag*nits contains 1000000, and the equal time of its perihelion September 16 1BOOK III.] OF NATURAL PHILOSOPHY. 50!16h . 10 . The places of this comet computed in this orbit by Mr. Bradley,and compared with the places observed by himself, his uncle Mr. Pound,and Dr. Halley, may be seen in the following table.From these examples it is abundantly evident that the motions of comets are no less accurately represented by our theory than the motions of theplanets commonly are by the theories of them ; and, therefore, by means ofthis theory, we may enumerate the orbits of comets, and so discover theperiodic time of a comet s revolution in any orbit; whence, at last, weshall have the transverse diameters of their elliptic orbits and their aphelion distances.That retrograde comet which appeared in the year 1607 described anorbit whose ascending; node (according to Dr. Halley s computation) was inb 20 2V ; arid the inclination of the plane of the orbit to the plane ofthe ecliptic 17 2 ;, whose perihelion was in ox 2 16 ; and its periheliondistance from the sun 58680 of such parts as the radius of the orbis mag--nns contains 100000; and the comet was in its perihelion October 16(l. 3".50 : which orbit agrees very nearly with the orbit of the comet which WHSseen in 1682. If these were not two cliiferent comets, but one and thesame, that comet will finish one revolution in the space of 75 years ; andthe greater axis of its orbit will be to the greater axis of the nrbis magimsas v/3 75 X 75 to 1, or as 1778 to 100, nearly. And the aphelion distance of this comet from the sun will be to the mean distance of the earthfrom the sun as about 35 to 1; from which data it will be no hard matterto determine the elliptic orbit of this comet. But these things are to besupposed on condition, that, after the space of 75 years, the same cometshall return again in the same orbit. The other comets seem to ascend togreater heights, and to require a longer time to perform their revolutions.But. because of the great number of comets, of the great distance of their502 THE MATHEMATICAL PRINCIPLES [BOOK IILaphelions from the sun, and of the slowness of their motions in the aphelions, they will, by their mutual gravitations, disturb each other ; so thattheir eccentricities arid the times of their revolutions will be sometimes alittle increased, and sometimes diminished. Therefore we are not to expect that the same comet will return exactly in the same orbit, and in thesame periodic times : it will be sufficient if we find the changes no greaterthan may arise from the causes just spoken of.And hense a reason may be assigned why comets are not comprehend-edwithin the limits of a zodiac, as the planets are; but, being confined to nobounds, are with various motions dispersed all over the heavens; namely,to this purpose, that in their aphelions, where their motions are exceedinglyslow, receding to greater distances one from another, they may suffer lessdisturbance from their mutual gravitations: and hence it is that the cometswhich descend the lowest, and therefore move the slowest in their aphelions,ought also to ascend the highest.The comet which appeared in the year 1GSO was in its perihelion lessdistant from the sun than by a sixth part of the sun s diameter; and because of its extreme velocity in that proximity to the sun, and some densityof the sun s atmosphere, it must have suffered some resistance and retardation ; and therefore, being attracted something nearer to the sun in evryrevolution, will at last fall down upon the body of the sun. Nay. in itsaphelion, where it moves the slowest, it may sometimes happen to be yetfarther retarded by the attractions of other comets, and in consequence ofthis retardation descend to the sun. So fixed stars, that have been gradually wasted by the light and vapours emitted from them for a long time,may be recruited by comets that fall upon them ; and from tlrs fresh supply of new fuel those old stars, acquiring new splendor, may pass for newstars. Of this kind are such fixed stars as appear on a sudden, and shinewith a wonderful brightness at first, and afterwards vanish by little andlittle. Such was that star which appeared in Cassiopeia s chair ; whichCornelius Gemma did not see upon the 8th of November, 1572, thoughhe was observing that part of the heavens upon that very night, and thesky was perfectly serene; but the next night (November 9) he saw itshining much brighter than any of the fixed stars, and scarcely inferior toVenus in splendor. Tycho Brake saw it upon the llth of the same month,when it shone with the greatest lustre; and from that time he observed itto decay by little and little; and in 16 months time it entirely disappeared. In the month of November, when it first appeared, its light wasequal to that of Venus. In the month of December its light was a littiediminished, and was now become equal to that of Jupiter. In January1573 it was less than Jupiter, and greater than Siriits ; and about theend of February and the beginning of March became equal to that star.In the months of April and May it was equal to a star of the second magHI.] OF NATURAL PHILOSOPHY. 503uitude; in June, July, and August, to a star of the third magnitude; inSeptember, October, and November, to those of the fourth magnitude; inDecember and January 1574 to those of the fifth;in February to thoseof the sixth magnitude; and in March it entirely vanished. Its colour atthe beginning was clear, bright, and inclining to white; afterwards ilturned a little yellow; and in March 1573 it became ruddy, like Mars orAlclebaran : in May it turned to a kind of dusky whiteness, like that weobserve in Saturn ; and that colour it retained ever after, but growing always more and more obscure. Such also was the star in the right foot oiSerpentarius, which Kepler s scholars first observed September 30, O.S.1604, with a light exceeding that of Jupiter, though the night before itwas not to be seen; and from that time it decreased by little and little,and in 15 or 16 months entirely disappeared. Such a new star appearingwith an unusual splendor is said to have moved Hipparchus to observe,and make a catalogue of, the fixed stars. As to those fixed stars that appear and disappear by turns, and increase slowly and by degrees, andscarcely ever exceed the stars of the third magnitude, they seem to be ofanother kind, which revolve about their axes, and, having a light and adark side, shew those two different sides by turns. The vapours whicharise from the sun, the fixed stars, and the tails of the comets, may meetat last with, and fall into, the atmospheres of the planets by their gravity,and there be condensed and turned into water and humid spirits; and fromthence, by a slow heat, pass gradually into the form of salts, and sulphurs,and tinctures, and mud, and clay, and sand, and stones, and coral, and otherterrestrial substances.GENERAL SCHOLIUM.The hypothesis of vortices is pressed with many difficulties. That everyplanet by a radius drawn to the sun may describe areas proportional to thetimes of description, the periodic times of the several parts of the vorticesshould observe the duplicate proportion of their distances from the sun ;but that the periodic times of the planets may obtain the sesquiplicate proportion of their distances from the sun; the periodic times of the parts ofthe vortex ought to be in the sesquiplicate proportion of their distances.That the smaller vortices may maintain their lesser revolutions aboutSaturn, Jupiter, and other planets, and swim quietly and undisturbed inthe greater vortex of the sun, the periodic times of the parts of the sun svortex should be equal ; but the rotation of the sun and planets about theiraxes, which ought to correspond with the motions of their vortices, recedefar from all these proportions. The motions of the comets are exceedinglyregular, are governed by the same laws with the motions of the planets,and can by no means be accounted for by the hypothesis of vortices ; forcomets are carried with very eccentric motions through all parts of the501 THE MATHEMATICAL PRINCIPLES [BOOK IILheavens indifferently, with a freedom that is incompatible with the notionof a vortex.Bodies projected in our air suffer no resistance but from the air. Withdraw the air, as is done in Mr. Boyle s vacuum, and the resistance ceases ;for in this void a bit of tine down and a piece of solid gold descend withequal velocity. Ajid the parity of reason must take place in the celestialspaces above the earth s atmosphere; in which spaces, where there is noair to resist their motions, all bodies will move with the greatest freedom;and the planets and comets will constantly pursue their revolutions in orbits given in kind and position, according to the laws above explained ; butthough these bodies may, indeed, persevere in their orbits by the mere lawsof gravity, yet they could by no means have at first derived the regularposition of the orbits themselves from those laws.The six primary planets are revolved about the sun in circles concentricwith the sun, and with motions directed towards the same parts, and almost in the same plane. Ten moons are revolved about the earth, Jupiterand Saturn, in circles concentric with them, wi h the same direction ofmotion, and nearly in the planes of the orbits of those planets ; but it isnot to be conceived that mere mechanical causes could give birth to somany regular motions, since the comets range over all parts of the heavensin very eccentric orbits; for by that kind of motion they pass easily throughthe orbs of the planets, and with great rapidity ; and in their aphelions,where they move the slowest, and are detained the longest, they recede tothe greatest distances from each other, and thence suffer the least disturbance from their mutual attractions. This most beautiful system of the sun,planets, and comets, could only proceed from the counsel and dominion of anintelligent and powerful Being. And if the fixed stars are the centres of other like systems, these, being formed by the like wise counsel, must be all subject to the dominion of One ; especially since the light of the fixed stars isof the same nature with the light of the sun, and from every system lightpasses into all the other systems : and lest the systems of the fixed starsshould, by their gravity, fall on each other mutually, he hath placed thosesystems at immense distances one from another.This Being governs all things, not as the soul of the world, but as Lordover all; and on account of his dominion he is wont to be called Lord God-rra TOKpaTup, or Universal Rider ; for God is a relative word, and has arespect to servants ; and Deity is the dominion of God not over his ownbody, as those imagine who fancy God to be the soul of the world, but overservants. The Supreme God is a Being eternal, infinite, absolutely perfect ; but a being, however perfect, without dominion, cannot be said to beLord God ; for we say, my God, your God, the God of Israel, the God ofGods, and Lord of Lords ; but we do not say, my Eternal, your Eternal.the Eternal of Israd} the Eternal of Gods; we do not say, my Infinite, o?{JOCK III.J Of NATURAL PHILOSOPHY. 505my Perfect : these are titles which have no respect to servants. The wordGod* usually signifies Lord ; but every lord is not a God. It is the dominion of a spiritual being which constitutes a God: a true, supreme, orimaginary dominion makes a true, supreme, or imaginary God And fromhis true dominion it follows that the true God is a living, intelligent, andpowerful Being ; and, from his other perfections, that he is supreme, ormost perfect. He is eternal and infinite, omnipotent and omniscient ; thatis, his duration reaches from eternity to eternity; his presence from infinityto infinity; he governs all things, and knows all things that are or can bedone. He is not eternity or infinity, but eternal and infinite; he is notduration or space, but he endures and is present. He endures for ever, andis every where present ; and by existing always and every where, he constitutes duration and space. Since every particle of space is always, andevery indivisible moment of duration is every where, certainly the Makerand Lord of all things cannot be never and no where. Every soul thathas perception is, though in different times and in different organs of senseand motion, still the same indivisible person. There are given successiveparts in duration, co-existent parts in space, but neither the one nor theother in the person of a man, or his thinking principle ; and much lesscan they be found in the thinking substance of God. Every man, so faras he is a thing that has perception, is one and the same man during hiswhole life, in all and each of his organs of sense. God is the same God,always and every where. He is omnipresent not virtually only, but alsosubstantially ; for virtue cannot subsist without substance. In himf areall things contained and moved; yet neither affects the other: God suffersnothing from the motion of bodies ; bodies find no resistance from the omnipresence of God. It is allowed by all that the Supreme God existsnecessarily ; and by the same necessity he exists always and every where.Whence also he is all similar, all eye, all ear, all brain, all arm, all powerto perceive, to understand, and to act; but in a manner not at all human,in a manner not at all corporeal, in a manner utterly unknown to us. Asa blind mail has no idea of colours, so have we no idea of the manner by* Dr. Pocock derives the Latin word Deus from the Arabic du (in the oblique case tit).which signifies Lord. And in this sense princes are called gods, Psal. Ixxxii. ver. 6; andJohn x. ver. 35. And Moses is called a god to his brother Aaron, and a god to Pharaoh,(Exod. iv. ver. 16 ; and vii. ver. 1). And in the same sense the souls of dead princes wereformerly, by the Heathens, culled gods, but falsely, because of their want of dominion.t This was the opinion of the Ancients. So Pythagoras, in Cicer. de Nat. Deor. lib. iThafes, Anaxagoras, Virgil, Georg. lib. iv. ver. 220; and ^Eneid, lib. vi. ver. 721. PhiloAllegor, at the beginning of lib. i. Aratu$, in his Phaenom. at the beginning. So also thesacred writers ; as St. Paul, Acts, xvii. ver 27, 28. St. John s Gosp. chap. xiv. ver. 2. Motet, in Dent. iv. ver. 39; and x ver. 14. David, Psal. cxxxix. ver. 7, 8, 9. Solomon, 1Kings, viii. ver. 27. Job, xxii. ver. 12, 13, 14. Jeremiah, xxiii. ver. 23, 24. The Idolaterssupposed the sun, moon, and stars, the souls of men, and other parts of the world, to beparts of the Supreme God, and therefore to be worshipped ; but erroneously.506 THE MATHEMATICAL PRINCIPLES [BOOK I1J.which the all-wise God perceives and understands all things. He is utterly void of all body and bodily figure, and can therefore neither l^e seen,nor heard, nor touched ; nor ought he to be worshipped under the representation of any corporeal thing. We have ideas of his attributes, butwhat the real substance of any thing is we know not. In bodies, we seeonly their figures and colours, we hear only the sounds, we touch only theiroutward surfaces, we smell only the smells, and taste the savours ; but theirinward substances are not to be known either by our senses, or by anyreflex act of our minds : much less, then, have we any idea of the substance of God. We know him only by his most wise and excellent contrivances of things, and final causes ; we admire him for his perfections ;but we reverence and adore him on account of his dominion : for we adorehim as his servants ; and a god without dominion, providence, and finalcauses, is nothing else but Fate and Nature. Blind metaphysical necessity, which is certainly the same always and every where, could produceno variety of things. All that diversity of natural things which we findsuited to different times and places could arise from nothing but the ideasand will of a Being necessarily existing. But, by way of allegory, Godis said to see, to speak, to laugh, to love, to hate, to desire, to give, to receive, to rejoice, to be angry, to fight, to frame, to work, to build ; for allour notions of God are taken from the ways of mankind by a certainsimilitude, which, though not perfect, has some likeness, however. Andthus much concerning God ; to discourse of whom from the appearancesof things, does certainly belong to Natural Philosophy.Hitherto we have explained the phenomena of the heavens and of oursea by the power of gravity, but have not yet assigned the cause of thispower. This is certain, that it must proceed from a cause that penetratesto the very centres of the sun and planets, without suffering the leastdiminution of its force; that operates not according to the quantity ofthe surfaces of the particles upon which it acts (as mechanical causes useto do), but according to the quantity of the solid matter which they contain,, and propagates its virtue on all sides to immense distances, decreasingalways in the duplicate proportion of the distances. Gravitation towardsthe sun is made up out of the gravitations towards the several particlesof which the body of the sun is composed ; and in receding from the sundecreases accurately in the duplicate proportion of the distances MS far asthe orb of Saturn, as evidently appears from the quiescence of the aphelions of the planets ; nay, and even to the remotest aphelions of the comets,if those aphelions are also quiescent. But hitherto I have not been ableto discover the cause of those properties of gravity from phenomena, andI frame no hypotheses ; for whatever is not deduced from the phenomenais to be called an hypothesis ; and hypotheses, whether metaphysical 01physical, whether of occult qualities or mechanical, have no place in exBOOK III.] OF NATURAL PHILOSOPHY. 507perimental philosophy. In this philosophy particular propositions areinferred from the phenomena, and afterwards rendered general by induction. Thus it was that the impenetrability, the mobility, and the impulsive force of bodies, and the laws of motion and of gravitation, werediscovered. And to us it is enough that gravity does really exist, and actaccording to the laws which we have explained, and abundantly serves toaccount for all the motions of the celestial bodies, and of our sea.And now we might add something concerning a certain most subtleSpirit which pervades and lies hid in all gross bodies ; by the force andaction of which Spirit the particles of bodies mutually attract one anotherat near distances, and cohere, if contiguous ; and electric bodies operate togreater distances, as well repelling as attracting the neighbouring corpuscles; and light is emitted, reflected, refracted, inflected, and heats bodies ;and all sensation is excited, and the members of animal bodies move at thecommand of the will, namely, by the vibrations of this Spirit, mutuallypropagated along the solid filaments of the nerves, from the outward organs of sense to the brain, and from the brain into the muscles. But theseare things that cannot be explained in few words, nor are we furnishedwith that sufficiency of experiments which is required to an accurate determination and demonstration of the laws by which this electric and elasticSpirit operates.END OP THE MATHEMATICAL P&LNCIPLE8.THE SYSTEM OF THE WORLD,THFSYSTEM OF THE WORLDIt. was the ancient opinion of not a few, in the earliest ages of philosophy, that the fixed stars stood immoveable in the highest parts of the world ;that, under the fixed stars the planets were carried about the sun ; that theearth, us one of the planets, described an annual course about the sun, whileby a diurnal motion it was in the mean time revolved about its own axis;and that the sun, as the common fire which served to warm the whole, wasfixed in the centre of the universe.This was the philosophy taught of old by Philolans, Aristarchus ofSantos, Plato in his riper years, and the whole sect of the Pythagoreans ;and this was the judgment of Anaximander, more ancient than any ofthem ; and of that wise Iring of the Rovnans, Numa Pompilins, who, asa symbol of the figure of the world with the sun in the centre, erected atemple in honour of Vesta, of .% i^und form, and ordained perpetual fire tobe kept in the middle of it.The Egyptians were early observers of the heavens ; and from them.,probably, this philosophy was spread abroad among other nations ; for fromthem it was, and the nations about them, that the Greeks, a people ofthemselves more addicted to the study of philology than of nature, derivedtheir first, as well as soundest, notions of philosophy ; and in the vestalceremonies we may yet trace the ancient spirit of the Egyptians ; for itwas their way to deliver their mysteries, that is, their philosophy of thingsabove the vulgar way of thinking, under the veil of religious rites andhieroglyphic symbols.It is not to be denied but that Anaxa&oras, Democritus, and others,did now and then start up, who would have it that the earth possessed thecentre of the world, and that the stars of all sorts were revolved towardsthe west about the earth quiescent in tk^ centre, some at a swifter, othersat a slower rate.However, it was agreed on both sides that the motions of the celestialbodies were performed in spaces altogether free and void of resistance. Thewhim of solid orbs was of a later date, introduced by Eudoxus, Calippus,and Aristotle; when the ancient philosophy began to decline, and to givenlace to the new prevailing fictions of the Greeks.But. above all things, the phenomena of comets can by no means consist612 THE SYSTEM OF THE WORLD.with the notion of solid orbs. The Chaldeans, the most learned astronomers of their time, looked upon the comets (which of ancient times beforehad been numbered among the celestial bodies) as a particular sort of planets, which, describing very eccentric orbits, presented themselves to our viewonly by turns, viz., once in a revolution, when they descended into thelower parts of their orbits.And as it was the unavoidable consequence of the hypothesis of solidorbs, while it prevailed, that the comets should be thrust down below themoon, so no sooner had the late observations of astronomers restored thecomets to their ancient places in the higher heavens, but these celestial spaceswere at once cleared of the incurnbrance of solid orbs, which by these observations were broke into pieces, and discarded for ever.Whence it was that the planets came to be retained within any certainbounds in these free spaces, and to be drawn off from the rectilinear courses,which, left to themselves, they should have pursued, into regular revolutions in curvilinear orbits, are questions which we do not know how theancients explained ; and probably it was to give some sort of satisfactionto this difficulty that solid orbs were introduced.The later philosophers pretend to account for it either by the action ofcertain vortices, as Kepler and Des Cartes ; or by some other principle ofimpulse or attraction, as Borelli, Honke, and others of our nation ; for,from the laws of motion, it is most certain that these effects must proceedfrom the action of some force or other.But our purpose is only to trace out the quantity and properties of thisforce from the phenomena (p. 218), and to apply what we discover in somesimple cases as principles, by which, in a mathematical way, we may estimate the effects thereof in more involved cases : for it would be endless andimpossible to bring every particular to direct and immediate observation.We said, in a mathematical way, to avoid all questions about the nature or quality of this force, which we would not be understood to determine by any hypothesis; and therefore call it by the general name of acentripetal force, as it is a force which is directed towards some centre ;and as it regards more particularly a body in that centre, we call it circumsolar, circum-terrestrial, circum-jovial ; and in like manner in respect ofother central bodies.That by means of centripetal forces the planets may be retained in certain orbits, we may easily understand, if we consider the motions of projectiles (p. 75, 76, 77) ; for a stone projected is by the pressure of its ownweight forced out of the rectilinear path, which by the projection alone itshould have pursued, and made to describe a curve line in the air; andthrough that crooked way is at last brought down to the ground ; and thegreater the velocity is with which it is projected, the farther it goes beforeit falls to the earth. We may therefore suppose the velocity to be so inTHE SYSTEM OF THE WORLD. 513creased, that it would describe an arc of 1, 2, 5, 10, 100. 1000 miles beforeit arrived at the earth, till at last, exceeding the limits of the earth, itshould pass quite by without touching it.Let AFB represent the surface of the earth, C its centre, VD, VE, VF,the curve lines which a body would describe, if projected in an horizontaldirection from the top of an high mountain successively "with more andmore velocity (p. 400) ; and, because the celestial motions are scarcely retarded by the little or no resistance of the spaces in which they are performed, to keep up the parity of cases, let us suppose either that there isno air about the earth, or at least that it is endowed with little or no powerof resisting ; and for the same reason tl a*: the body projected with a lessvelocity describes the lesser arc VD, and with a greater velocity the greaterarc VE. and, augmenting the velocity, it goes farther and farther to F andG, if the velocity was still more and more augmented, it would reach atlast quite beyond the circumference of the earth, and return to the mountain from which it was projected.And since the areas which by this motion it describes by a radius drawn