Principia Mathematica (Motte Translation, 1848) — part 30 of 45

It has been, now of a long time, observed by others, that all sorts of heavy bodies (allowance being made for the inequality of retardation which they suffer from a small power of resistance in the air) descend to the earth from equal heights in equal times ; and that equality of times we may distinguish to a great accuracy, by the help of pendulums. I tried the thing in gold, silver, lead, glass, sand, common 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 centre of oscillation of the other. The boxes hanging by equal threads of 11 feet made a couple of pendulums perfectly equal in weight and figure, and equally receiving the resistance of the air. And, placing the one by the other, I observed them to play together forward and backward, for a long time, with equal vibrations. And therefore the quantity of matter in the gold (by Cor. 1 and 6, Prop. XXIY, Book II) was to the quantity of mat- ter in the wood as the action of the motive force (or vis motrix) upon all the gold to the action of the same upon all the wood ; that is, as the weight of the one to the weight of the other : and the like happened in the other bodies. By these experiments, in bodies of the same weight, I could man- ifestly have discovered a difference of matter less than the thousandth part of the whole, had any such been. But, without all doubt, the nature of gravity towards the planets is the same as towards the earth. For, should we imagine our terrestrial bodies removed to the orb of the moon, and there, together with the moon, deprived of all motion, to be let go, so as to fall together towards the earth, it is certain, .from what we have demonstra- ted before, that, in equal times, they would describe equal spaces with the moon, and of consequence are to the moon, in quantity of matter, as their weights to its weight. Moreover, since the satellites of Jupiter perform

Book III.] of natural philosophy. 395

their revolutions in times which observe the sesquiplicate proportion of their distances from Jupiter's centre, their accelerative gravities towards Jupiter will be reciprocally as the squares of their distances from Jupiter's centre; that is, equal, at equal distances. And, therefore, these satellites, if supposed to fall towards Jupiter from equal heights, would describe equal spaces 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 be let fall at equal distances from the sun, they would, in their descent towards the sun, describe equal spaces in equal times. But forces which equally accelerate unequal bodies must be as those bodies : that is to say, the weights of the planets towards the sun must be as their quantities of matter. Further, that the weights of Jupiter and of his satellites towards the sun are proportional to the several quantities of their matter, appears from the exceedingly regular motions of the satellites (by Cor. 3, Prop. LXY, Book I). For if some of those bodies were more strongly attracted to the sun in proportion to their quantity of matter than others, the motions of the sat- ellites would be disturbed by that inequality of attraction (by Cor. 2, Prop. LXV, Book I). If, at equal distances from the sun, any satellite, in pro- portion to the quantity of its matter, did gravitate towards the sun with a force greater than Jupiter in proportion to his, according to any given pro- portion, suppose of c? to e / then the distance between the centres of the sun and of the satellite's orbit would be always greater than the distance be- tween the centres of the sun and of Jupiter nearly in the subduplicate of that proportion : as by some computations I have found. And if the sat- ellite did gravitate towards the sun with a force, lesser in the proportion of e to dj the distance of the centre of the satellite's orb from the sun would be less than the distance of the centre of Jupiter from the sun in the subdu- plicate of the same proportion. Therefore if, at equal distances from the sun, the accelerative gravity of any satellite towards the sun were greater or less than the accelerative gravity of Jupiter towards the sun but by one joVo- part of the whole gravity, the distance of the centre of the satellite's orbit from the sun would be greater or less than the distance of Jupiter from the sun by one 2oV^ P^^^ ^^ ^^ whole distance; that is, by a iifih part of the distance of the utmost satellite from the centre of Jupiter ; an eccentricity of the orbit which would be very sensible. But the orbits of the satellites are concentric to Jupiter, and therefore the accelerative gravities of Jupiter, and of all its satellites towards the sun, are equal among themselves. And by the same argument, the weights of Saturn and of his satellites towards the sun, at equal distances from the sun, are as their several quantities of matter ; and the weights of the moon and of the earth towards the sun are either none, or accurately proportional to the masses of matter which they contain. iBut some they are, by Cor. 1 and 3, Prop. Y.

But further ; the weights of all the parts of every planet towards any other

396 THE MATHEMATICAL PRINCIPLES [BoOK III.

planet are one to -another as the matter in the several parts ; for if some parts 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 most abounds, would gravitate more or less than in proportion to the quantity of matter in the whole. Nor is it of any moment whether these parts are external or internal ; for if, for example, we should imagine the terrestrial bodies with us to be raised up to the orb of the moon, to be there compared with its body : if the weights of such bodies were to the weights of the ex- ternal parts of the moon as the quantities of matter in the one and in the other respectively ; but to the weights of the internal parts in a greater or less proportion, then likewise the weights of those bodies would be to the weight of the whole moon in a greater or less proportion; against what we have shewed above.

Cor. 1. Hence the weights of bodies do not depend upon their forms and textures ; for if the weights could be altered with the forms, they would 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 the earth ; 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 the quality of all bodies within the reach of our experiments ; and therefore (by Rule III) to be affirmed of all bodies whatsoever. If the mther, or any other body, were either altogether void of gravity, or were to gravitate less in proportion to its quantity of matter, then, because (according to Aris- totle, Des Cartes, and others) there is no difference betwixt that and other bodies but in mere form of matter, by a successive change from form to form, it might be changed at last into a body of the same condition with those which gravitate most in proportion to their quantity of matter ; and, on the other hand, the heaviest bodies, acquiring the first form of that body, might by degrees quite lose their gravity. And therefore the weights would depend upon the forms of bodies, and with those forms might be changed : contrary to what was proved in the preceding Corollary.

CoR. 3. All spaces are not equally full : for if all spaces were equally full, 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 short of the specific gravity of quicksilver, or gold, or any other the most dense body ; and, therefore, neither gold, nor any other body, could descend in air ; for bodies do not descend in fluids, unless they are specifically heavier than the fluids. And if the quantity of matter in a given space can, by any rarefaction, be diminished, what should hinder a diminution to infinity ?

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 vacuum must be granted.

Book III.] of natural philosophy. 397

By bodies of the same density, I mean those whose vires inertim are in the proportion of their bulks.

Cor. 5. The power of gravity is of a different nature from the power of magnetism ; for the magnetic attraction is not as the matter attracted. Some bodies are attracted more by the magnet ; others less ; most bodies not at all. The power of magnetism in one and the same body may be increased and diminished ; and is sometimes far stronger, for the quantity of matter, than the power of gravity ; and in receding from the magnet decreases not in the duplicate but almost in the triplicate proportion of the distance, as nearly as I could judge from some rude observations.

PROPOSITION yil. THEOREM YII.

That there is a poioer of gravity tending to all bodies, proportional to

the several quantities of matter tohich they co7itain.

That all the planets mutually gravitate one towards another, we have proved 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 from the centre of the planet. And thence (by Prop. LXIX, Book I, and its Corollaries) it follows, that the gravity tending towards all the planets is proportional to the matter which they contain.

Moreover, since all the parts of any planet A gravitate towards any other planet B ; and the gravity of every part is to the gravity of the whole as the matter of the part to the matter of the whole ; and (by Law III) to every action corresponds an equal re-action ; therefore the planet B will, 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 the whole 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 arises from, and is compounded of, the forces of gravity towards all its parts. Magnetic and electric attractions afford us examples of this ; for all at- traction towards the whole arises from the attractions towards the several parts. The thing may be easily understood in gravity, if we consider a greater planet, as formed of a number of lesser planets, meeting together in one globe ; for hence it ivoidd appear that the force of the whole must arise from the forces of the component parts. If it is objected, that, ac- cording to this law, all bodies with us must mutually gravitate one to- wards another, whereas no such gravitation any where appears, I answer, that since the gravitation towards these bodies is to the gravitation towards the whole earth as these bodies are to the whole earth, the gravitation to- wards 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 any body is reciprocally as the square of the distance of places from the parti- cles ; as appears from Cor. 3, Prop. LXXIY, Book I.

398 THE MATHEMATICAL PRINCIPLES [BoOK III.

PHOPOSITION VIII. THEOREM YIII.

In tivo spheres mutually gravitating each towards the other, if the matter in places on all sides round about and equi-distant from the centres is similar, the weight of either sphere toivards the other will be recipro- cally as the square of the distance betiveen their cetitres. After I had found that the force of gravity towards a whole planet did arise from and was compounded of the forces of gravity towards all its parts, and towards every one part w^as in the reciprocal proportion of the squares of the distances from the part, I was yet in doubt whether that re- ciprocal duplicate proportion did accurately hold, or but nearly so, in the total force compounded of so many partial ones; for it might be that the proportion which accurately enough took place in greater distances should be wide of the truth near the surface of the planet, where the distances of the particles are unequal, and their situation dissimilar. But by the help of Prop. LXXV and LXXVI, Book I, and their Corollaries, I was at last satisfied of the truth of the Proposition, as it now lies before us.

Cor. 1. Hence we may find and compare together the weights of bodies towards different planets ; for the weights of bodies revolving in circles about planets are (by Cor. 2, Prop. IV, Book I) as the diameters of the circles directly, and the squares of their periodic times reciprocally ; and their weights at the surfaces of the planets, or at any other distances from their centres, are (by this Prop.) greater or less in the reciprocal duplicate proportion of the distances. Thus from the periodic times of Venus, re- volving about the sun, in 224*^. 16f^, of the utmost circumjovial satellite revolving about Jupiter, in 16'. 16 -?/. ; of the Huygenian satellite about Saturn in 15^. 22 1\ ; and of the moon about the earth in 27'\ T\ 43' ; compared with the mean distance of Venus from the sun, and with the greatest heliocentric elongations of the outmost circumjovial satellite from Jupiter's centre, 8' 16"; of the Huygenian satellite from the centre of Saturn, 3' 4" ; and of the moon from the earth, 10' 33" : by computa- tion I found that the weight of equal bodies, at equal distances from the centres 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, toVtj 3"oVtj ^^^ T6 ¥2¥2 respectively. Then because as the distances are increased or di- minished, the weights are diminished or increased in a duplicate ratio, the weights 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 their very superficies, will be as 10000, 943, 529, and 435 respectively. How much the weights of bodies are at the superficies of the moon, will be shewn hereafter.

CoR. 2. Hence likewise we discover the quantity of matter in the several

Book III.] of natural philosophy. 399

planets ; for their quantities of matter are as the forces of gravity at equal distances from their centres ; that is, in the sun, Jupiter, Saturn, and the earth, as 1, t^Vt; aoVn ^^^ T69V¥2 respectively. If the parallax of the sun be taken greater or less than 10" 30'", the quantity of matter in the earth must be augmented or diminished in the triplicate of that pro- portion.

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 similar spheres are, at the surfaces of those spheres, as the diameters of the spheres j and therefore the densities of dissimilar spheres are as those weights applied to the diameters of the spheres. But the true diameters of the Sun, Jupi- ter, Saturn, and the earth, were one to another as 10000, 997, 791, and 109; and the weights towards the same as 10000, 943, 529, and 435 re- spectively ; and therefore their densities are as 100, 94i, 67, and 400. The density of the earth, which comes out by this computation, does not depend upon the parallax of the sun, but is determined by the parallax of the moon, and therefore is here truly defined. The sun, therefore, is a little denser than Jupiter, and Jupiter than Saturn, and the earth four times denser than the sun ; for the sun, by its great heat, is kept in a sort of a rarefied state. The moon is denser than the earth, as shall appear after- ward.

Cor. 4. The smaller the planets are, they are, cceteris paribus, of so much the greater density ; for so the powers of gravity on their several surfaces come nearer to equality. They are likewise, ccateris parihis, of the greater density, as they are nearer to the sun. So Jupiter is more dense than Saturn, and the earth than Jupiter ; for the planets were to be placed at different distances from the sun, that, according to their degrees of 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 con- verted into ice, and in the orb of Mercury would quickly fly away in va- pour ; for the light of the sun, to which its heat is proportional, is seven times denser in the orb of Mercury than with us : and by the thermometer I have found that a sevenfold heat of our summer sun will make water boil. Nor are we to doubt that the matter of Mercury is adapted to its heat, and is therefore more dense than the matter of our earth : since, in a denser matter, the operations of Nature require a stronger heat.

PROPOSITION IX. THEOREM IX.

That the force of gravity, considered doiumvard from the surface of the planets, decreases uearly in the proportion of the distances from their centres.

If the matter of the planet were of an uniform density, this Proposi- tion would be accurately true (by Prop. LXXIII, Book I). The error,

400 THE MATHEMATICAL PRINCIPLES [BoOK III.

tlierefore, can be no greater than what may arise from the inequality of the density.

PROPOSITION X. THEOREM X.

That tJie motions of the planets in the heavens may subsist an exceedingly

long time.

In the Scholium of Prop. XL, Book II, I have shewed that a globe of water frozen into ice, and moving freely in our air, in the time that it would describe the length of its semi-diameter, would lose by the resistance of the air 4 ^Vg part of its motion ; and the same proportion holds nearly in all globes, how great soever, and moved with whatever velocity. But that our globe of earth is of greater density than it would be if the whole consisted of water only, I thus make out. If the whole cujisisted of water only, whatever was of less density than water, because of its less specific gravity, would emerge and float above. And upon this account, if a globe of terrestrial matter, covered on all sides with water, was less dense than water, it would emerge somewhere ; and, the subsiding water falling back, would be gathered to the opposite side. And such is the condition of our earth, which in a great measure is covered with seas. The earth, if it was not for its greater density, would emerge from the seas, and, accord- ing 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 same argument, the spots of the sun, which float upon the lucid matter thereof, are lighter than that matter ; and, however the planets have been formed while they were yet in fluid masses, all the heavier matter subsided to the centre. Since, therefore, the common matter of our earth on the surface thereof is about twice as heavy as water, and a little lower, in mines, is found about three, or four, or even five times more heavy, it is probable that the quantity of the whole maj^ter of the earth may be five or six times greater than if it consisted all of water ; especially since I have before shewed that the earth is about four times more dense than Jupiter. Ifj therefore, Jupiter is a little more dense than water, in the space of thirty days, in which that planet describes the length of 459 of its semi-diame- ters, it would, in a medium of the same density with our air, lose almost a tenth part of its motion. But since the resistance of mediums decreases in proportion to their weight or density, so that water, which is 13| times lighter than quicksilver, resists less in that proportion ; and air, which is 860 times lighter than water, resists less in the same proportion ; therefore in the heavens, where the weight of the medium in which the planets move is immensely diminished, the resistance will almost vanish.

It is shewn in the Scholium of Prop. XXII, Book II, that at the height of 200 miles above the earth the air is more rare than it is at the super- ficies of the earth in the ratio of 30 to 0,0000000000003998, or as

Book III.] of natural philosophy. 401

75000000000000 to 1 nearly. And hence the planet Jupiter, revolving in a medium of the same density with that superior air, would not lose by the resistance of the medium the 1000000th part of its motion in 1000000 years. In the spaces near the earth the resistance is produced only by the air, exhalations, and vapours. When these are carefully exhausted by the air-pump from under the receiver, heavy bodies fall within the receiver with perfect freedom, and without the least sensible resistance: gold itself, and the lightest down, let fall together, will descend with equal velocity ; and though they fall through a space of four, six, and eight feet, they will come to the bottom at the same time; as appears from experiments. And there- fore the celestial regions being perfectly void of air and exhalations, the planets and comets meeting no sensible resistance in those spaces will con- tinue 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 from hence follow.

PROPOSITION XL THEOPvEM XI.

That the common centre of gravity of the earth, the sun, and all the planets, is i?n?novable. For (by Cor. 4 of the Laws) that centre either is at rest, or moves uni- formly forward in a right line ; but if that centre moved, the centre of the world would move also, against the Hypothesis.

^ PROPOSITION XIL THEOREM XH.

That the sun is agitated by a perpetual m^otion, but never recedes far from the common centre of gravity of all the planets. For since (by Cor. 2, Prop. VIII) the quantity of matter in the sun is to the quantity of matter in Jupiter as 1067 to 1 ; and the distance of Jupi- ter from the sun is to the semi-diameter of the sun in a proportion but a small matter greater, the common centre of gravity of Jupiter and the sun will fall upon a point a little without the surface of the sun. By the same argument, since the quantity of matter in the sun is to the quantity of matter in Saturn as 3021 to 1, and the distance of Saturn from the sun is to 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 point a little within the surface of the sun. And, pursuing the principles of this computation, we should find that though the earth and all the planets were placed on one side of the sun, the distance of the common centre of gravity of all from the centre of the sun would scarcely amount to one diameter of

26

402 THE MATHEMATICAL PBINCIPLES [BoOK III.

the sun. In other cases, the distances of those centres are always less ; and therefore, since that centre of gravity is in perpetual rest, the sun, accord- ing to the various positions of the planets, must perpetually be moved every way, but will never recede far from that centre.

Cor. Hence the common centre of gravity of the earth, the sun, and all the 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, and are therefore, according to their powers of gravity, in perpetual agitation, as the Laws of Motion require, it is plain that their moveable centres can- not be taken for the immovable centre of the world. If that body were to be placed in the centre, towards which other bodies gravitate most (accord- ing to common opinion), that privilege ought to be allowed to the sun ; but since the sun itself is moved, a fixed point is to be chosen from which the centre of the sun recedes least, and from which it would recede yet less if the body of the sun were denser and greater, and therefore less apt to be moved.

PROPOSITION XIII. THEOREM XIII.

The planets 7?iovein ellipses which have their common focus in the centre of the snn ; and, by radii drawn to that centre, they describe areas pro- portional to tJie times of description.

We have discoursed above of these motions from the Phasnomena. Now that we know the principles on which tney depend, from those principles we deduce the motions of the heavens d priori. Because the weights of the planets towards the sun are reciprocally as the squares of their distan- ces from the sun's centre, if the sun was at rest, and the other planets did not mutually act one upon another, their orbits would be ellipses, having the sun in their common focus ; and they would describe areas proportional to the times of description, by Prop, I and XI, and Cor. 1, Prop. XIII, Book I. But the mutual actions of the planets one upon another are so very small, that they may be neglected ; and by Prop. LXVI, Book I, they less disturb the motions of the planets around the sun in motion than if those motions were performed about the sun at rest.

It is true, that the action of Jupiter upon Saturn is not to be neglected ; for the force of gravity towards Jupiter is to the force of gravity towards the sun (at equal distances, Cor. 2, Prop. VIII) as 1 to 1067 ; and therefore in the conjunction of Jupiter and Saturn, because the distance of Saturn from Jupiter is to the distance of Saturn from the sun almost as 4 to 9, the gravity of Saturn towards Jupiter will be to the gravity of Saturn towards the sun as 81 to 16 X 1067; or, as 1 to about 2\ 1. And hence arises a perturbation of the orb of Saturn in every conjunction of this planet with Jupiter, so sensible, that astronomers are puzzled with it. As the planet

Book 111.] of natukal philosophy. 403

is differently situated in these conjunctions, its eccentricity is sometimes augmented, sometimes diminished ; its aphelion is sometimes carried for- ward, sometimes backward, and its mean motion is by turns accelerated and retarded; yet the whole error in its motion about the sun, though arising from so great a force, may be almost avoided (except in the mean motion) by placing the lower focus of its orbit in the common centre of gravity of Jupiter and the sun (according to Prop. LXVII, Book I), and therefore that error, when it is greatest, scarcely exceeds two minutes ; and the greatest error in the mean motion scarcely exceeds two minutes yearly. But in the conjunction of Jupiter and Saturn, the accelerative forces of gravity of the sun towards Saturn, of Jupiter towards Saturn, and of Jupiter towards the

sun, are almost as 16, SI, and -^ ;0r 156609; and therefore

the difference of the forces of gravity of the sun towards Saturn, and of Jupiter towards Saturn, is to the force of gravity of Jupiter towards the sun as 65 to 156609, or as 1 to 2409. But the greatest power of Saturn to disturb the motion of Jupiter is proportional to this difierence ; and therefore the perturbation of the orbit of Jupiter is much less than that of Saturn's. The perturbations of the other orbits are yet far less, except that the orbit of the earth is sensibly disturbed by the moon. The common centre of gravity of the earth and moon moves in an ellipsis about the sun in the focus thereof, and, by a radius drawn to the sun, describes areas pro- portional to the times of description. But the earth in the mean time by a menstrual motion is revolved about this common centre.

PROPOSITION XIV. THEOREM XIV.

The aphelions and nodes of the orbits of the planets are fixed»

The aphelions are immovable by Prop. XI, Book I ; and so are the planes of the orbits, by Prop. I of the same Book. And if the planes are fixed, the nodes must be so too. It is true, that some inequalities may arise from the mutual actions of the planets and comets in their revolu- tions ; but these will be so small, that they may be here passed by. -

CoR. 1. The fixed stars are immovable, seeing they keep the same posi- tion to the aphelions and nodes of the planets.

Cor. 2. And since these stars are liable to no sensible parallax from the annual motion of the earth, they can have no force, because of their im- mense distance, to produce any sensible effect in our system. Not to mention that the fixed stars, every where promiscuously dispersed in the heavens, by their contrary attractions destroy their mutual actions, by Prop. LXX, Book I.

SCHOLIUM.

Since the planets near the sun (viz. Mercury, Venus, the Earth, and

404 THE MATHEMATICAL PRINCIPLES [BoOK III.

Mars) are so small that they can act with but little force upon each other, therefore their aphelions and nodes must be fixed, excepting in so far as they are disturbed by the actions of Jupiter and Saturn, and other higher bodies. And hence we may find, by the theory of gravity, that tlieir aphe- lions move a little in consequentia, in respect of the fixed stars, and that in the sesquiplicate proportion of their several distances from the sun. So that if the aphelion of Mars, in the space of a hundred years, is carried 33' 20" in consequentia, in respect of the fixed stars, the aphelions of the Earth, of Venus, and of Mercury, will in a hundred years be carried for- wards 17' 40", 10' 53", and 4' 16", respectively. But these motions are so inconsiderable, that we have neglected them in this Proposition.

PROPOSITION XY. PROBLEM I.

To find the principal diameters of the orbits of the planets. They are to be taken in the sub-sesquiplicate proportion of the periodic times, by Prop. XV, Book I, and then to be severally augmented in the proportion of the sum of the masses of matter in the sun and each planet to the first of two mean proportionals betwixt that sum and the quantity of matter in the sun, by Prop. LX, Book I.

PROPOSITION XVI. PROBLEM 11.

To find the eccentricities and aphelions of the planets. This Problem is resolved by Prop. XVIII, Book I.

PROPOSITION XVII. THEOREM XV.

That the dinrnal motions of the planets are uniform, and that the libration of the moon arises from its diurnal motion. The Proposition is proved from the first Law of Motion, and Cor. 22, Prop. LXVI, Book I. Jupiter, with respect to the fixed stars, revolves in 9\ 56' ; Mars in 24\ 39' ; Venus in about 23^ ; the Earth in 23^ 5^' ; the Sun in 25 i days, and the moon in 27 days, 7 hours, 43'. These things appear by the Ph^enomena. The spots in the sun's body return to the same situation on the sun's disk, with respect to the earth, in 27 \ days ; and therefore with respect to the fixed stars the sun revolves in about 25^ days. But because the lunar day, arising from its uniform revolution about its axis, is menstrual, that is, equal to the time of its periodic revolution in its orb, therefore the same face of the moon will be always nearly turned to the upper focus of its orb ; but, as the situation of that focus requires, will deviate a little to one side and to the other from the earth in the lower focus ; and this is the libration in longitude ; for the libration in latitude arises from the moon^s latitude, and the inclination of its axis to the plane of the ecliptic. This theory of the libration of the moon, Mr. N. Mercator,

Book III.] of natural philosophy. 405

in his Astronomy, published at the beginning of the year 1676, explained more fully out of the letters I sent him. The utmost satellite of Saturn seems to revolve about its axis with a motion like this of the moon, respect- ing Saturn continually with the same face ; for in its revolution round Saturn, as often as it comes to the eastern part of its orbit, it is scarcely visible, and generally quite disappears ; which is like to be occasioned by some spots in that part of its body, which is then turned towards the earth, as M. Cassini has observed. So also the utmost satellite of Jupiter seems to revolve about its axis with a like motion, because in that part of its body which is turned from Jupiter it has a spot, which always appears as if it were in Jupiter's own body, whenever the satellite passes between Jupiter and our eye.

PROPOSITION XYIII. THEOREM X.YI.

That the axes of the pla?iets are less than the diameters draiun perpen-

dlcidar to the axes. The equal gravitation of the parts on all sides would give a spherical figure to the planets, if it was not for their diurnal revolution in a circle. By that circular motion it comes to pass that the parts receding from the axis endeavour to ascend about the equator ; and therefore if the matter is in a fluid state, by its ascent towards the equator it will enlarge the di- ameters there, and by its descent towards the poles it will shorten the axis. So the diameter of Jupiter (by the concurring observations of astronomers) is found shorter betwixt pole and pole than from east to west. And, by the same argument, if our earth was not higher about the equator than at the poles, the seas would subside about the poles, and, rising towards the equator, would lay all things there under water.

PROPOSITION XIX. PROBLEM III.

To find the proportion of the axis of a planet to the diameters per pen-