Principia Mathematica (Motte Translation, 1848) — part 39 of 45

That retrograde comet which appeared in the year 1607 described an orbit whose ascending node (according to Dr. Halley'' s computation) was in ^ 20° 21' ; and the inclination of the plane of the orbit to the plane of the ecliptic 17° 2' ; whose perihelion was in ox 2° 16' ; and its perihelion distance from the sun 58680 of such parts as the radius of the or6i5 mag- nus contains 100000 ; and the comet was in its perihelion October 16''. 3^. 50' ; which orbit agrees very nearly with the orbit of the comet which was seen in 1682. If these were not two different comets, but one and the same, that comet will finish one revolution in the space of 75 years ; and the greater axis of its orbit will be to the greater axis of the orbis magnus as v/^ : 75 X 75 to 1, or as 1778 to 100, nearly. And the aphelion dis- tance of this comet from the sun will be to the mean distance of the earth from the sun as about 35 to 1 ; from which data it will be no hard matter to determine the elliptic orbit of this comet. But these things are to be supposed on condition, that, after the space of 75 years, the same comet shall return again in the same orbit. The other comets seem to ascend to greater heights, and to require a longer time to perform their revolutions.

But, because of the great number of comets, of the great distance of their

,502 THE MATHEMATICAL PRINCIPLES [BoOK IBL

aphelions from the sun, and of the slowness of their motions in the aphe- lions, they will, by their mutual gravitations, disturb each other ; so that their eccentricities and the times of their revolutions will be sometimes a little increased, and sometimes diminished. Therefore we are not to ex- pect that the same comet will return exactly in the same orbit, and in the same periodic times : it will be sufficient if we find the changes no greater than may arise from the causes just spoken of.

And hence a reason may be assigned why comets are not comprehended within the limits of a zodiac, as the planets are ; but, being confined to no bounds, are with various motions dispersed all over the heavens ; namely, to this purpose, that in their aphelions, where their motions are exceedingly slow, receding to greater distances one from another, they may suffer less disturbance from their mutual gravitations : and hence it is that the comets which descend the lowest, and therefore move the slowest in their aphelions, ought also to ascend the highest.

The comet which appeared in the year 1680 was in its perihelion less distant from the sun than by a sixth part of the sun's diameter : and be- cause of its extreme velocity in that proximity to the sun, and some density of the sun's atmosphere, it must have suffered some resistance and retarda- tion ; and therefore, being attracted something nearer to the sun in every revolution, will at last fall down upon the body of the sun. Nay, in its aphelion, where it moves the slowest, it may sometimes happen to be yet farther retarded by the attractions of other comets, and in consequence of this retardation descend to the sun. So fixed stars, that have been gradu- ally wasted by the light and vapours emitted from them for a long time, may be recruited by comets that fall upon them ; and from this fresh sup- ply of new fuel those old stars, acquiring new splendor, may pass for new stars. Of this kind are such fixed stars as appear on a sudden, and shine with a wonderful brightness at first, and afterwards vanish by little and little. Such was that star which appeared in Cassiopeia^ s chair ; which Cornelius Gemma did not see upon the 8th of November , 1572, though he was observing that part of the heavens upon that very night, and the sky was perfectly serene; but the next night {November 9) he saw it shining much brighter than any of the fixed stars, and scarcely inferior to Venus in splendor. Tycho Brake saw it upon the 11th of the same month, when it shone with the greatest lustre ; and from that time he observed it to decay by little and little ; and in 16 months' time it entirely disap- peared. In the month of November^ when it first appeared, its light was equal to that of Venus. In the month of December its light was a little diminished, and was now become equal to that of Jupiter. In January 1573 it was less than Jupiter, and greater than Sirius ; and about the end 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 mag-

Book III.] of natural philosophy. 503

nitude; in June, July, and August, to a star of the third magnitude; in September, October, and November, to those of the fourth magnitude ; in December and January 1574 to those of the fifth ; in February to those of the sixth magnitude ; and in March it entirely vanished. Its colour at the beginning was clear, bright, and inclining to "vvhite; afterwards it turned a little yellow; and in March 1573 it became ruddy, like Mar^ ox Aldebaran : in May it turned to a kind of dusky whiteness, like that W€ observe in Saturn ; and that colour it retained ever after, but growing al- ways more and more obscure. Such also was the star in the right foot of Serpentarius, which Keplefs scholars first observed September 30, O.S. 1604, with a light exceeding that of Jupiter, though the night before it was 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 appearing with an unusual splendor is said to have moved Ilipparchus to observe, and make a catalogue of, the fixed stars. As to those fixed stars that ap- pear and disappear by turns, and increase slowly and by degrees, and scarcely ever exceed the stars of the third magnitude, they seem to be of another kind, which revolve about their axes, and, having a light and a dark side, shew those two different sides by turns. The vapours which arise from the sun, the fixed stars, and the tails of the comets, may meet at 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 from thence, 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 other terrestrial substances.

GENERAL SCHOLIUM.

The hypothesis of vortices is pressed with many difficulties. That every planet by a radius drawn to the sun may describe areas proportional to the times of description, the periodic times of the several parts of the vortices should observe the duplicate proportion of their distances from the sun ; but that the periodic times of the planets may obtain the sesquiplicate pro- portion of their distances from the sun, the periodic times of the parts of the vortex ought to be in the sesquiplicate proportion of their distances. That the smaller vortices may maintain their lesser revolutions about- Saturn, Jupiter, and other planets, and swim quietly and undisturbed in the greater vortex of the sun, the periodic times of the parts of the sun's vortex should be equal ; but the rotation of the sun and planets about their axes, which ought to correspond with the motions of their vortices, recede far from all these proportions. The motions of the comets are exceedingly regular, 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 ; for comets are carried with very eccentric motions through all parts of the

504 THE MATHEMATICAL PRINCIPLES [BoOK IIL

heavens indifferently, with a freedom that is incompatible with the notion of a vortex.

Bodies projected in our air suffer no resistance but from the air. With- draw the air, as is done in Mr, Boyle's vacuum, and the resistance ceases ; for in this void a bit of fine down and a piece of solid gold descend with egual velocity. And the parity of reason must take place in the celestial spaces above the earth's atmosphere ; in which spaces, where there is no air to resist their motions, all bodies will move with the greatest freedom ; and the planets and comets will constantly pursue their revolutions in or- bits given in kind and position, according to the laws above explained ; but though these bodies may, indeed, persevere in their orbits by the mere laws of gravity, yet they could by no means have at first derived the regular position of the orbits themselves from those laws.

The six primary planets are revolved about the sun in circles concentric with the sun, and with motions directed towards the same parts, and al- most in the same plane. Ten moons are revolved about the earth, Jupiter and Saturn, in circles concentric with them, with the same direction of motion, and nearly in the planes of the orbits of those planets ; but it is not to be conceived that mere mechanical causes could give birth to so many regular motions, since the comets range over all parts of the heavens in very eccentric orbits ; for by that kind of motion they pass easily through the orbs of the planets, and with great rapidity ; and in their aphelions, where they move the slowest, and are detained the longest, they recede to the greatest distances from each other, and thence suffer the least disturb- ance from their mutual attractions. This most beautiful system of tlie sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centres of oth- er like systems, these, being formed by the like wise counsel, must be all sub- ject to the dominion of One ; especially since the light of the fixed stars is of the same nature with the light of the sun, and from every system light passes into all the other systems : and lest the systems of the fixed stars should, by their gravity, fall on each other mutually, he hath placed those systems at immense distances one from another.

This Being governs all things, not as the soul of the world, but as Lord over all ; and on account of his dominion he is wonfc to be called Lord God navTOKpdruyp, or Universal Ruler ; for God is a relative word, and has a respect to servants ; and Deity is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely per- fect ; but a being, however perfect, without dominion, cannot be said to be Lord God ; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords ; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods j we do not say, my Infinite, or

Book III.] of natural philosophy. 505

my Perfect : these are titles which have no respect to servants. The word God* usually signifies Lord; but every lord is not a God. It is the do- minion of a spiritual being which constitutes a God : a true, supreme, or imaginary dominion makes a true, supreme, or imaginary God. And from his true dominion it follows that the true God is a living, intelligent, and powerful Being ; and, from his other perfections, that he is supreme, or most perfect. He is eternal and infinite, omnipotent and omniscient ; that is, his duration reaches from eternity to eternity ; his presence from infinity to infinity ; he governs all things, and knows all things that are or can be done. He is not eternity or infinity, but eternal and infinite ; he is not duration or space, but he endures and is present. He endures for ever, and is every where present ; and by existing always and every where, he consti- tutes duration and space. Since every particle of space is always, and every indivisible moment of duration is every where, certainly the Maker and Lord of all things cannot be never and no lohere. Every soul that has perception is, though in different times and in different organs of sense and motion, still the same indivisible person. There are given successive parts in duration, co-existent parts in space, but neither the one nor the other in the person of a man, or his thinking principle ; and much less can they be found in the thinking substance of God. Every man, so far as he is a thing that has perception, is one and the same man during his whole 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 also substantially ; for virtue cannot subsist without substance. In himf are all things contained and moved ; yet neither affects the other : God suffers nothing from the motion of bodies ; bodies find no resistance from the om- nipresence of God. It is allowed by all that the Supreme God exists necessarily ; 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 power to 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. As a blind man 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 di), which signifies Lord. And in this sense princes are called gods, Psal. Ixxxii. ver. 6 ; and John 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 were formerly, by the Heathens, called gods, but falsely, because of their want of dominion.

t This was the opinion of the Ancients. So Pythagoras, in Cicer. de Nat. Dear. lib. i. Thales, Anaxagoras, Virgil^ Georg. lib. iv. ver. 220; and iEneid, lib. vi. ver. 721. Philo Allegor, at the beginning of lib. i. Aratus, in his Phaenom. at the beginning. So also the sacred writers ; as St. Paul, Acts, xvii. ver. 27, 28. St. John's Gosp. chap. xiv. ver. 2. Mo- ses, in Deut. iv. ver. 39; and x. ver. 14. David, Psal. cxxxix. ver. 7, 8, 9. Solomon, 1 Kings, viii. ver. 27. Job, xxii. ver. 12, 13, 14. Jeremiah, xxiii. ver. 23, 24. The Idolaters supposed the sun, moon, and stars, the souls of men, and other parts of the world, to be parts of the Supreme God, and therefore to be worshipped ; but erroneously.

506 • THE MATHEMATICAL PRINCIPLES [BoOK III.

which the all-wise God perceives and understands all things. He is ut- terly void of all body and bodily figure, and can therefore neither be seen, nor heard, nor touched ; nor ought he to be worshipped under the repre- sentation of any corporeal thing. We have ideas of his attributes, but what the real substance of any thing is we know not. In bodies, we see only their figures and colours, we hear only the sounds, we touch only their outward surfaces, we smell only the smells, and taste the savours ; but their inward substances are not to be known either by our senses, or by any reflex act of our minds : much less, then, have we any idea of the sub- stance of God. We know him only by his most wise and excellent con- trivances of things, and final causes ; we admire him for his perfections ; but we reverence and adore him on account of his dominion : for we adore him as his servants ; and a god without dominion, providence, and final causes, is nothing else but Fate and Nature. Blind metaphysical neces- sity, which is certainly the same always and every where, could produce no variety of things. All that diversity of natural things which we find suited to different times and places could arise from nothing but the ideas and will of a Being necessarily existing. But, by way of allegory, God is said to see, to speak, to laugh, to love, to hate, to desire, to give, to re- ceive, to rejoice, to be angry, to fight, to frame, to work, to build ; for all our notions of God are taken from the ways of mankind by a certain similitude, which, though not perfect, has some likeness, however. And thus much concerning God ; to discourse of whom from the appearances of things, does certainly belong to Natural Philosophy.

Hitherto we have explained the pheenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. This is certain, that it must proceed from a cause that penetrates to the very centres of the sun and planets, without suffering the least diminution of its ^force ; that operates not according to the quantity of the surfaces of the particles upon which it acts (as mechanical causes use to do), but according to the quantity of the solid matter which they con- tain, and propagates its virtue on all sides to immense distances, decreasing always in the duplicate proportion of the distances. Gravitation towards the sun is made up out of the gravitations towards the several particles of which the body of the sun is composed ; and in receding from the sun decreases accurately in the duplicate proportion of the distances as far as the orb of Saturn, as evidently appears from the quiescence of the aphe- lions 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 able to discover the cause of those properties of gravity from phaenomena, and I frame no hypotheses ; for whatever is not deduced from the ph^enomena is to be called an hypothesis ; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in ex-

Book IILJ of natural philosophy. 507

perimental philosophy. In this philosophy particular propositions are inferred from the phaenomena, and afterwards rendered general by induc- tion. Thus it was that the impenetrabilityj the mobility, and the impul- sive force of bodies, and the laws of motion and of gravitation, were discovered. And to us it is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our sea.

And now we might add something concerning a certain most subtle Spirit which pervades and lies hid in all gross bodies ; by the force and action of which Spirit the particles of bodies mutually attract one another at near distances, and cohere, if contiguous ; and electric bodies operate to greater distances, as well repelling as attracting the neighbouring corpus- cles ; and light is emitted, reflected, refracted, inflected, and heats bodies ; and all sensation is excited, and the members of animal bodies move at the command of the will, namely, by the vibrations of this Spirit, mutually propagated along the solid filaments of the nerves, from the outward or- gans of sense to the brain, and from the brain into the muscles. But these are things that cannot be explained in few words, nor are we furnished with that sufficiency of experiments which is required to an accurate deter- mination and demonstration of the laws by which this electric and elastic Spirit operates.

END OF THE MATHEMATICAL PRINCIPLES.

THE SYSTEM OF THE WOKLD.

THE

SYSTEM OF THE ¥OELD

It was the ancient opinion of not a few, in the earliest ages of philoso- phy, 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 the earth, as one of the planets, described an annual course about the sun, while by 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, was fixed in the centre of the universe.

This was the philosophy taught of old by Philolmis, Aristarchus of Samos, Plato in his riper years, and the whole sect of the Pythagoreans ; and this was the judgment of Anaximander, more ancient than any of them ; and of that wise king of the Romans, Numa Pompilius, who, as a symbol of the figure of the world with the sun in the centre, erected a temple in honour of Vesta, of a round form, and ordained perpetual fire to be 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 from them it was, and the nations about them, that the Greeks, a people of themselves more addicted to the study of philology than of nature, derived their first, as well as soundest, notions of philosophy ; and in the vestal ceremonies we may yet trace the ancient spirit of the Egyptians ; for it was their way to deliver their mysteries, that is, their philosophy of things above the vulgar way of thinking, under the veil of religious rites and hieroglyphic symbols.

It is not to be denied but that Anaxagoras, Democritus, and others, did now and then start up, who would have it that the earth possessed the centre of the world, and that the stars of all sorts were revolved towards the west about the earth quiescent in the centre, some at a swifter, others at a slower rate.

However, it was agreed on both sides that the motions of the celestial bodies were performed in spaces altogether free and void of resistance. The whim of solid orbs was of a later date, introduced by Eudoxus, CalippuSy and Aristotle ; when the ancient philosophy began to decline, and to give place to the new prevailing fictions of the Greeks.

But, above all things, the phaenomena of comets can by no means consist

512 THE SYSTEM OF THE WORLD.

with the notion of solid orbs. The Chaldeans, the most learned astrono- mers of their time, looked upon the comets (which of ancient times before had been numbered among the celestial bodies) as a particular sort of plan- ets, which, describing very eccentric orbits, presented themselves to our view only by turns, viz., once in a revolution, when they descended into the lower parts of their orbits.

And as it was the unavoidable consequence of the hypothesis of solid orbs, while it prevailed, that the comets should be thrust down below the moon, so no sooner had the late observations of astronomers restored the comets to their ancient places in the higher heavens, but these celestial spaces were at once cleared of the incumbrance of solid orbs, which by these ob- servations were broke into pieces, and discarded for ever.

Whence it was that the planets came to be retained within any certain bounds in these free spaces, and to be drawn off from the rectilinear courses, which, left to themselves, they should have pursued, into regular revolu- tions in curvilinear orbits, are questions which we do not know how the ancients explained ; and probably it was to give some sort of satisfaction to this difficulty that solid orbs were introduced.

The later philosophers pretend to account for it either by the action of certain vortices, as Kepler and Des Cartes ; or by some other principle of impulse or attraction, as Borelli, Hooke, and others of our nation ; for, from the laws of motion, it is most certain that these effects must proceed from the action of some force or other.

But our purpose is only to trace out the quantity and properties of this force from the phaenomena (p. 218), and to apply what we discover in some simple cases as principles, by which, in a mathematical way, we may esti- mate the effects thereof in more involved cases ; for it would be endless and impossible to bring every particular to direct and immediate observation.

We said, in a mathematical loay, to avoid all questions about the na- ture or quality of this force, which we would not be understood to deter- mine by any hypothesis ; and therefore call it by the general name of a centripetal 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 circum- solar, circum-terrestrial, circum-jovial ; and in like manner in respect of other central bodies.

That by means of centripetal forces the planets may be retained in cer- tain orbits, we may easily understand, if we consider the motions of pro- jectiles (p. 75, 76, 77) ; for a stone projected is by the pressure of its own weight forced out of the rectilinear path, which by the projection alone it should have pursued, and made to describe a curve line in the air ; and through that crooked way is at last brought down to the ground ; and the greater the velocity is with which it is projected, the farther it goes before it falls to the earth. We may therefore suppose the velocity to be so in-

THE SYSTEM OF THE WORLD.

513

creased, that it would describe an arc of 1, 2, 5, 10, 100, 1000 miles before it arrived at tbe earth, till at last, exceeding the limits of the earth, it should pass quite by without touching it.

Let AFB represent the surface of the earth, C its centre, YD, VE, VF, the curve lines which a body would describe, if projected in an horizontal direction from the top of an high mountain successively with more and

more velocity (p. 400) ; and, because the celestial motions are scarcely re- tarded by the little or no resistance of the spaces in which they are per- formed, to keep up the parity of cases, let us suppose either that there is no air about the earth, or at least that it is endowed with little or no power of resisting; and for the same reason that the body projected with a less velocity describes the lesser arc VD, and with a greater velocity the greater arc YE, and, augmenting the velocity, it goes farther and farther to F and G, if the velocity was still more and more augmented, it would reach at last quite beyond the circumference of the earth, and return to the moun- tain from which it was projected.

And since the areas which by this motion it describes by a radius drawn to the centre of the earth are (by Prop. 1, Book 1, Princip. Math.) propor- tional to the times in which they are described, its velocity, when it returns to the mountain, will be no less than it was at first ; and, retaining the- same velocity, it will describe the same curve over and over, by the same law.

514 THE SYSTEM OF THE WORLD.

But if we now imagine bodies to be projected in the directions of lines parallel to the horizon from greater heights, as of 5, 10, 100, 1000, or more miles, or rather as many semi-diameters of the earth, those bodies, accord- ing to their different velocity, and the diiferent force of gravity in different heights, will describe arcs either concentric with the earth, or variously eccentric, 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 per- pendicular direction, the perpetual deflection thereof towards the earth from the right line in which it was projected is a proof of its gravitation to the earth, no less certain than its direct descent when only suffered to fall freely from rest ; so the deviation of bodies moving in free spaces from rectilinear paths, and perpetual deflection therefrom towards any place, is a sure indication of the existence of some force which from all quarters impels those bodies towards that place.

And as, from the supposed existence of gravity, it necessarily follows that all bodies about the earth must press downwards, and therefore must either descend directly to the earth, if they are let fall from rest, or at least perpetually deviate from right lines towards the earth, if they are projected obliquely ; so from the supposed existence of a force directed to any centre, it will follow, by the like necessity, that all bodies upon which this force acts must either descend directly to that centre, or at least devi- ate perpetually tow^ards it from right lines, if otherwise they should have moved obliquely in these right lines.

And how from the motions given we may infer the forces, or from the forces given we may determine the motions, is shewn in the two first Books of our Principles of Philosophy.

If the earth is supposed to stand still, and the fixed stars to be revolved in free spaces in the space of 24 hours, it is certain the forces by which the fixed stars are retained in their orbs are not directed to the earth, but to 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 the fi^ed stars describe areas exactly proportional to jjie times of description. Then, because the periodic times are equal (by Cor. Ill, Prop. lY, 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 like consequences may be drawn from the supposed diurnal motion of the planets.

That forces should be directed to no body on which they physically de- pend, but to innumerable imaginary points in the axis of the earth, is an hypothesis too incongruous. It is more incongruous still that those forces should increase exactly in proportion of the distances from this axis : for

THE SYSTEM OF THE WORLD. 515

this is an indication of an increase to immensity, or rather to infinity ; whereas the forces of natural things commonly decrease in receding from the fountain from which they flow. But, what is yet more absurd, neither are the areas described by the same star proportional to the times, nor are its revolutions performed in the same orb ; for as the star recedes from the neighbouring pole, both areas and orb increase ; and from the increase of the area it is demonstrated that the forces are not directed to the axis of the earth. And this difficulty (Cor. 1, Prop. II) arises from the twofold motion that is observed in the fixed stars, one diurnal round the axis of the earth, the other exceedingly slow round the axis of the ecliptic. And the explication thereof requires a composition of forces so perplexed and so variable, that it is hardly to be reconciled with any physical theory.

That there are centripetal forces actually directed to the bodies of the sun, 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 axe described, as is evident from its velocity compared with its apparent diame- ter ; for its motion is slower when its diameter is less (and therefore its distance greater), and its motion is swifter when its diameter is greater.

The revolutions of the satellites of Jupiter about that planet are more regular (p. 386) ; for they describe circles concentric with Jupiter by equa- ble motions, as exactly as our senses can distinguish.

And so the satellites of Saturn are revolved about this planet with mo- tions nearly (p. 387) circular and equable, scarcely disturbed by any eccen- tricity hitherto observed.

That Yenus and Mercury are revolved about the sun, is demonstrable from their moon-like appearances (p. 388) : when they shine with a full face, they are in those parts of their orbs which in respect of the earth lie beyond the sun ; when they appear half full, they are in those parts which lie over against the sun ; when horned, in those parts which lie between the earth and the sun ; and sometimes they pass over the sun's disk, when directly interposed between the earth and the sun.

And Venus, with a motion almost uniform, describes an orb nearly cir- cular and concentric with the sun.

But Mercury, with a more eccentric motion, makes remarkable ap- proaches to the sun, and goes off again by turns ; but it is always swifter as it is near to the sun, and therefore by a radius drawn to the sun still describes areas proportional to the times.

Lastly, that the earth describes about the sun, or the sun about the earth, by a radius from the one to the other, areas exactly proportional to the times, is demonstrable from the apparent diameter of the sun com- pared with its apparent motion.

These are astronomical experiments ; from which it follows, by Prop. I,

516

THE SYSTEM OF THE WORLD.

II, III, in the first Book of our Principles^ and their Corollaries (p. 212, 213, 214), that there are centripetal forces actually directed (either accu- rately or without considerable error) to the centres of the earth, of Jupi- ter, of Saturn, and of the sun. In Mercury, Venus, Mars, and the lesser planets, where experiments are wanting, the arguments from analogy must be allowed in their place.

That those forces (p. 212, 213, 214) decrease in the duplicate propor- tion 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 are one to another (p. 3S6, 3S?) in the sesquiplicate proportion of their dis- tances 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 the micrometer, and by the eclipses of the satellites, wrote to me, that it holds to all the accuracy that possibly can be discerned by our senses. And he sent me the dimensions of their orbits taken by the micrometer, and re- duced to the mean distance of Jupiter from the earth, or from the sun, tosrether with the times of their revolutions, as follows : —

The greatest elon- gation of the sate- lites from the cen- tre of Jupiter as seen from the sun.

1st 1 48 or 108 2d 3 01 or 181 3d 4 46 or 286 4th 8 13i or 493i

The periodic times of their revohitions.

d h ' "

1 18 28 36

3 13 17 54

7 03 59 36

16 18 5 13

Whence the sesquiplicate proportion may be easily seen. For example ; the 16'^. 18^ 05' 13" is to the time 1^. IS^ 28' 36" as 493i" x V 493^" to lOS" X / lOS", neglecting those small fractions which, in observing, cannot be certainly determined.

Before the invention of the micrometer, the same distances were deter- mined in semi-diameters of Jupiter thus : —

Distance of the 1st

2d

3d

4th

By Galileo. ... 6

10

16

28

" Simon Marius . 6

10

16

26