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BS, meeting BC in C; and at the end of the fecond part of the time, the body, by cor 1. of the laws, will be found in C, in the fame plane with the triangle ASB. Join SC; and because SB and e C are parallel, the triangle SBC will be equal to the triangle SCD, and therefore also to the triangle SAB. By the like argument, if the centripetal force acts fucceffively in C, D, E, &c. and makes the body in each fingle particle of time to defcribe the right lines CD, DE, EF, &c. they will all lie in the fame plane; and the triangle SCD will be equal to the triangle SBC, and SDE to SCD, and SEF to SDE. And therefore, in equal times, equal areas are described in one immoveable plane; and by compofition, any fums SADS, SAFS, of those areas are, one to the other, as the times in which they are defcribed. Now, let the number of thofe triangles be augmented, and their fize dimished in infinitum; and then, by the preceding lemmas, their ultimate perimeter ADF will be a curve line: and therefore the centripetal force by which the body is perpetually drawn back from the tangent of this curve will act continually; and any defcribed areas SADS, SAFS, which are always proportional to the times of defcription, will, in this cafe alfo, be proportional to those times Q. E. D.

COR. 1. The velocity of a body attracted towards an immoveable centre, in spaces void of refiftances, is reciprocally as the perpendicular let fall from that centre on the right line which touches the orbit. For the velocitics in thefe places A, B, C, D, E, are as the bafes AB, BC, DE, EF, of equal triangles; and these bases are reciprocally as the perpendiculars let fall upon them.

COR 2. If the chords AB, BC, of two arcs fucceffively defcribed in equal times by the fame body, in spaces void of refiftance, are completed into a parallelogram ABCV, and the diagonal BV of this parallelogram, in the pofition which it ultimately acquires when those arcs are diminished in infinitum, is produced both ways, it will pafs through the centre of force.

COR. 3. If the chords AB, BC, and DE, EF, of arcs described in equal times, in spaces void of refiftance, are completed into the parallelograms ABCV, DEFZ, the forces in B and E are one to the other in the ultimate ratio of the diagonals BV, EZ, when those arcs are diminished in infinitum. For the motions BC and EF of the body (by cor. 1. of the laws), are compounded of the motions Be, BV and Ef, EZ; but BV and EZ, which are equal to Ce and Ff, in the demonftration of this propofition, were generated by the impulfes of the centripetal force in B and E, and are therefore proportional to thofe impulfes.

COR. 4. The forces by which bodies, in spaces void of refiftance, are drawn back from rectilinear motions, and turned into curvilinear orbits, are one to another as the verfed fines of arcs defcribed in equal times; which verfed fines tend to the centre of force, and bifect the chords when thefe arcs are diminished to infinity. For fuch verfed fines are the halves of the diagonals mentioned in cor. 3.

COR. 5. And therefore thofe forces are to the force of gravity, as the faid verfed fines to the

verfed fines perpendicular to the horizon of thofe parabolic arcs which projectiles describe in the fame time.

COR. 6. And the fame things do all hold good (by cor. 5. of the laws) when the planes in which the bodies are moved, together with the centres of force, which are placed in thofe planes, are not at reft, but move uniformly forward in right lines. THEOR. II. Every body that moves in any curve line defcribed in a plane, and, by a radius drawn to a point either immoveable or moving forward with an unform rectilinear motion, describes about that point areas proportional to the times, is urged by the centripetal force directed to that point.

CASE I. For every body that moves in a curve line is (by law 1.) turned afide from its rectilinear courfe by the action of some force that impels it; and that force by which the body is turned off from its rectilinear course, and made to defcribe in equal times the least equal triangles SAB, SBC, SCD, &c. about the immoveable point S, (by Prop. 40. E. 1. and law 2.) acts in the place B according to the direction of a line parallel to C; that is, in the direction of the line BS; and in the place C according to the direction of a line parallel to dD, that is, in the direction of the line CS, &c.; and therefore acts always in the direction of lines tending to the immoveable point S. Q. E. D.

CASE II. And (by cor. 5. of the laws) it is indifferent whether the fuperficies in which a body defcribes a curvilinear figure be quiefcent, or moves together with the body, the figure defcribed, and its point S, uniformly forward in right lines.

COR. I. In non-refifting spaces of mediums, if the areas are not proportional to the times, the forces are not directed to the point in which the radii meet; but deviate therefrom in confequentia, or towards the parts to which the motion is directed, if the description of the areas is accelerated; but in antecedentia if retarded.

COR. 2. And even in refifting mediums, if the defcription of the areas is accelerated, the direc tions of the forces deviate from the point in which the radii meet, towards the parts to which the motion tends.

SCHOLIUM. A body may be urged by a centripetal force compounded of several forces. In which cafe the meaning of the propofition is, that the force which results out of all tends to the point S. But if any force acts perpetually in the direction of lines perpendicular to the described furface, this force will make the body to deviate from the plane of its motion, but will neither augment nor diminish the quantity of the described furface; and is therefore not to be neglected in the compofition of forces.

THEOR. III. Every body that, by a radius drawn to the centre of another body, howsoever moved, describes areas about that centre proportional to the times, is urged by a force compounded of the centripetal forces tending to that other body, and of all the accelerative force by which the other body is impelled.-The demonftration of this is a natural confequence of the theorem immediately preceding.

Hence, if the one body L, by a radius drawn to the other body T, defcribes areas proportional to

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the times, and from the whole force by which the COR. 5. If the periodic times are as the radii, firft body L is urged (whether that force is fimple, and therefore the velocities equal, the centripetal or, according to cor. 2. of the laws, compounded forces will be reciprocally as the radii; and the of feveral forces), we fubdu&t that whole accele-, contrary. rative force by which the other body is urged; the whole remaining force by which the firft body is urged will tend to the other body T, as its centre. And vice verfa, if the remaining force tends nearly to the other body T, thofe areas will be nearly proportional to the times.

If the body L, by a radius drawn to the other body T, defcribes areas, which compared with the times are very unequal, and that other body T be either at reft, or moves uniformly forward in a right line, the action of the centripetal force tending to that other body T is either none at all, or it is mixed and combined with very powerful actions of other forces: and the whole force compounded of them all, if they are many, is directed to another (immoveable or moveable) centre. The fame thing obtains when the other body is actuated by any other motion whatever; provided that centripetal force is taken which remains after fubducting that whole force acting upon that other body T.

SCHOLIUM. Because the equable defcription of areas indicate that a centre is refpected by that force with which the body is most affected, and by which it is drawn back from its rectilinear motion, and retained in its orbit, we may always be allowed to use the equable defcription of areas as an indication of a centre about which all circular motion is performed in free spaces.

THEOR. IV. The centripetal forces of bodies which by equable motions defcribe different circles, tend to the centres of the fame circles; and are one to the other as the fquares of the arcs defcribed in equal times applied to the radii of circles. For thefe forces tend to the centres of the circles, (by theor 2. and cor. 2. theor. 1.) and are to one another as the verfed fines of the leaft arcs described in equal times, (by cor. 4. theor 1.) that is as the squares of the fame arcs applied to the diameters of the circles, by one of the lemmas: and therefore fince thofe arcs are as arcs described in any equal times, and the diameters are as the radii, the forces will be as the fquares of any arcs described in the fame time, applied to the radii of the circles. Q. E. D.

COR. I. Therefore, since those arcs are as the velocities of the bodies, the centripetal forces are in a ratio compounded of the duplicate ratio of the velocities directly, and of the fimple ratio of the radii inversely.

COR. 2. And fince the periodic times are in a ratio compounded of the ratio of the radii direct ly, and the ratio of the velocities inverfely; the centripetal forces are in a ratio compounded of the ratio of the radii directly, and the duplicate ratio of the periodic times inversely.

COR. 3. Whence, if the periodic times are equal, and the velocities therefore as the radii, the centripetal forces will be alfo as the radii; and the contrary.

COR. 4. If the periodic times and the velocities are both in the subduplicate ratio of the radii, the centripetal forces will be equal among themselves; and the contrary.

COR. 6. If the periodic times are in the fefquiplicate ratio of the radii, and therefore the velocities reciprocally in the fubduplicate ratio of the radii, the centripetal forces will be in the duplicate ratio of the radii inverse; and the contrary.

COR. 7. And univerfally, if the periodic time is as any power Rn of the radius R, and therefore the velocity reciprocally as the power Rn-1 of the radius, the centripetal force will be reciprocally as the power R2n−2 of the radius; and the contrary.

COR. 8. The fame things all hold concerning the times, the velocities, and forces, by which bodies defcribe the fimilar parts of any fimilar figures, that have their centres in a fimilar pofition within those figures, as appears by applying the demonftrations of the preceding cafes to those. And the application is easy, by only fubstituting the equable defcription of areas in the place of equable motion, and ufing the diftances of the bodies from the centres inftead of the radii.

COR. 9. From the fame demonftration it likewife follows, that the arc which a body, uniformly revolving in a circle by means of a given centripetal force, describes in any time, is a mean proportional between the diameter of the circle and the space which the fame body, falling by the fame given force, would defcend through in the fame given time.

"By means of the preceding propofition and its corollaries (fays Sir ISAAC), we may discover the proportion of a centripetal force to any other known force, fuch as that of gravity. For if a body by means of its gravity revolves in a circle concentric to the earth, this gravity is the centripetal force of that body. But from the defcent of heavy bodies, the time of one entire revolution, as well as the arc described in any given time, is given (by cor. 9. of this theorem). And by fuch propo fitions Mr HUYGENS, in his excellent book De Ho rologio Ofcillatorio, has compared the force of gravity with the centrifugal forces of revolving bodies.

The preceding propofition may alfo be demonftrated in the following manner:-In any circle fuppofe a polygon to be infcribed of any number of fides. And if a body, moved with a given velocity along the fides of the polygon, is reflected from the circle at the feveral angular points; the force with which, at every reflection it ftrikes the circle, will be as its velocity: and therefore the fum of the forces, in a given time, will be as that velocity and the number of reflections conjunctly; that is, (if the fpecies of the polygon be given), as the length defcribed in that given time, and increafed or diminished in the ratio of the fame length to the radius of the circle; that is, as the fquare of that length applied to the radius; and therefore, if the polygon, by having its fides diminifhed in infinitum, coincides with the circle, as the fquare of the arc defcribed in a given time applied to the radius. This is the centrifugal force, with which the body impels the circle; and to which the contrary force, wherewith the circle conti nually repels the body towards the centre, is equal.

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On these principles hangs the whole of Sir Ifaac the earth about the fun, are in the fefquiplicate Newton's mathematical philofophy. He now proportion of their mean distances from the fun. shows how to find the centre to which the forces 5. The primary planets, by radii drawn to the impelling any body are directed, having the velo-earth, defcribe areas no way sproportionable to city of the body given and finds the centrifugal the times: but the areas which they defcribe by force to be always as the verfed fine of the naf- radii drawn to the fun are proportional to the cent arc directly, and as the fquare of the time in- times of defcription. 6. The moon, by a radius verfely; or directly as the fquare of the velocity, drawn to the centre of the earth, describes an area and inverfely as the chord of the nafcent arc. proportional to the time of defcription. All these From these premifes he deduces the method of phenomena are undeniable from aftronomical obfinding the centripetal force directed to any given fervations, and are explained at large under the point when the body revolves in a circle; and article ASTRONOMY. The mathematical demon. this whether the central point is near or at an im- ftrations are next applied by Sir Ifaac Newton in menfe diftance; fo that all the lines drawn from it the following propofitions: may be taken for parallels. The fame thing he fhows with regard to bodies revolving in fpirals, ellipfes, hyperbolas, or parabolas.-Having the figures of the orbits given, he shows alfo how to find the velocities and moving powers; and, in fhort, folves all the most difficult problems relating to the celestial bodies with an aftonishing degree of mathematical fkill. These problems and demonstrations are all contained in the firft book of the Principia; to which we must refer those who with for farther information.

SECT. VI. RULES for PHILOSOPHICAL REA

SONING.

In his 2d book Sir ISAAC treats of the properties of fluids, and their powers of refiftance; and lays down fuch principles as entirely overthrow the doctrine of DES CARTES's vortices, which was the fashionable system in his time. In the 3d book, he begins particularly to treat of the natural phenomena, and apply them to the mathematical principles formerly demonftrated; and, as a neceffary preliminary to this part, he lays down the following rules for reasoning in natural philofophy:

1. We are to admit no more caufes of natural things than fuch as are both true and fufficient to explain their natural appearances.

2. Therefore to the fame natural effects we must always affign, as far as poffible, the fame causes. 3. The qualities of bodies which admit neither intenfion nor remiffion of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be efteemed the univerfal qualities of all bodies whatsoever.

4. In EXPERIMENTAL PHILOSOPHY, we are to look upon propofitions collected by general induction from phenomena as accurately or very nearly true, notwithstanding any contrary hypothefes that may be imagined, till fuch time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions.

The phenomena first confidered, are, 1. That the fatellites of Jupiter, by radii drawn to the centre of their primary, defcribe areas proportional to the times of the defcription; and that their periodic times, the fixed ftars being at reft, are in the fefquiplicate ratio of their diftances from its centre. 2. The fame thing is likewife obferved of the phenomena of Saturn. 3. The five primary planets, Mercury, Venus, Mars, Jupiter, and Saturn, with their several orbits, encompass the fun. 4. The fixed ftars being fuppofed at reft, the pcnodic times of the five primary planets, and of VOL. XVI. PART I.

PROP. I. The forces by which the satellites of Jupiter are continually drawn off from rectilinear motions, and retained in their proper orbits, tend to the centre of that planet; and are reciprocally as the fquares of the diftances of those fatellites from that centre. The former part of this propofition appears from theor. 2. or 3. and the latter from cor. 6. of theor. 5. and the fame thing we are to understand of the fatellites of Saturn.

PROP. II. The forces by which the primary planets are continually drawn off from rectilinear motions, and retained in their proper orbits, tend to the fun; and are reciprocally as the fquares of the diftances from the fun's centre. The former part of this propofition is manifeft from phenomenon 5. juft mentioned, and from theor. 2.; the latter from the phenomenon 4. and cor. 6. of theor 4. But this part of the propofition is with great accuracy deducible from the quiefcence of the aphelion points. For a very small aberration from the reciprocal duplicate proportion would produce a motion of the apfides, fenfible in every fingle revolution, and in many of them enormourly great.

PROP. III. The force by which the moon is retained in its orbit, tends towards the earth; and is reciprocally as the fquare of the distance of its place from the centre of the earth. The former part of this propofition is evident from phenom. 5. and theor. 2. the latter from phenom. 6. and theor. 2. or 3. It is alfo evident from the very flow motion of the moon's apogee; which, in every single revolution, amounting but to 3° 3′ in confequentia, may be neglected: and this more fully appears from the next propofition.

PROP. IV. The moon gravitates towards the earth, and by the force of gravity is continually drawn off from a rectilinear motion, and tetained in its orbit. The mean diftance of the moon from the earth in the fyzigies in femidiameters of the latter, is about 604. Let us affume the mean diftance of 60 femidiameters in the fyzigies; and fuppofe one revolution of the moon in refpect of the fixed ftars to be completed in 27d. 7h. 43', as aftronomers have determined; and the circumference of the earth to amount to 123,249,600 Paris feet. Now, if we imagine the moon, deprived of all her motion, to be let go, so as to defcend towards the earth with the impulse of all that force by which it is retained in its orbit, it will, in the fpace of one minute of time, defcribe in its fall 15 Paris feet. For the verfed fine of that arc which the moon, in the space of one minute of time, describes by its mean motion at the

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felves, they will (by rules 1, and 2.) have the fame caufe. And therefore the force which retains the moon in its orbit, is that very force which we commonly call gravity; becaufe otherwife, this little moon at the top of a mountain muft either be without gravity, or fall twice as fwiftly as heavy bodies ufe to do.

distance of 60 femidiameters of the earth, is nearly 15 Paris feet; or, more accurately, 15 feet I inch and one line. Wherefore, fince that force, in approaching to the earth, increafes in the reciprocal duplicate proportion of the diftance; and, upon that account, at the furface of the earth is 60 X 60 times greater than at the moon; a body in our regions, falling with that Having thus demonftrated that the moon is reforce ought, in the fpace of one minute of time, tained in its orbit by its gravitation towards the to describe 60X60X15 Paris feet; and in the earth, it is easy to apply the fame demonftration space of one fecond of time to defcribe 15 of to the motions of the other fecondary planets, thofe feet; or, more accurately, 15 feet 1 inch, and of the primary planets round the fun, and line. And with this very force we actually thus to fhow that gravitation prevails throughout find that bodies here on earth do really defcend. the whole creation. After which Sir Ifaac proFor a pendulum ofcillating feconds in the latitude ceeds to fhow from the fame principles, that the of Paris, will be three Paris feet and 84 lines in heavenly bodies gravitate towards each other, and length, as Mr HUYGENS has obferved. And the contain different quantities of matter, or have dif fpace which a heavy body defcribes by falling one ferent denfities in proportion to their bulks. fecond of time, is to half the length of the pendulum in the duplicate ratio of the circumference of the circle to its diameter; and is therefore 15 Paris feet, 1 inch, 1 line. And therefore the force by which the moon is retained in its orbit, becomes, at the very furface of the earth, equal to the force of gravity which we obferve in heavy bodies there. And therefore (by rules 1, and 2.) the force by which the moon is retained in its orbit is that very fame force which we commonly call GRAVITY. For were gravity another force different from that, then bodies defcending to the earth with the joint impulfe of both forces, would fall with a double velocity, and, in the fpace of one fecond of time, would defcribe 30% Paris feet; altogether against experience.

The demonftration of this propofition may be more diffufely explained after the following manner. Suppofe feveral moons to revolve about the earth, as in the fyftem of Jupiter or Saturn, the periodic times of thofe moons would (by the argument of induction) obferve the fame law which Kepler found to obtain among the planets; and therefore their centripetal forces would be reci. procally as the fquares of the diftances from the centre of the earth, by Prop. I. Now, if the lowest of these were very fmall, and were fo near the earth as almost to touch the tops of the higheft mountains, the centripetal force thereof, retaining it in its orbit, would be very nearly equal to the weights of any terreftrial bodies that fhould be found upon the tops of these mountains; as may be known from the foregoing calculation. Therefore, if the fame little moon fhould be deferted by its centrifugal force that carries it through its orbit, it would defcend to the earth; and that with the fame velocity as heavy bodies do actually defcend with upon the tops of thofe very mountains, because of the equality of forces that obliges them both to defcend. And if the force by which that loweft moon would defcend were different from that of gravity, and if that moon were to gravitate towards the earth, as we find terreftrial bodies do on the tops of mountains, it would then defcend with twice the velocity, as being impelled by both these forces confpiring together. Therefore, fince both thefe forces, that is, the gravity of heavy bodies, and the centripetal forces of the moons, refpect the centre of the earth, and are fimilar and equal between them

PROP. V. All bodies gravitate towards every planet; and the weight of bodies towards the fame planet, at equal distances from its centre are proportional to the quantities of matter they contain.

It has been confirmed by many experiments, that all forts of heavy bodies (allowance being made for the inequality of retardation by fome fmall refiftance of the air) defcend to the earth from equal heights in equal times; and that equa lity of times we may distinguish to a great accuracy by the help of pendulums. Sir Ifaac Newton tried the thing in gold, filver, lead, glass, fand, common falt, wood, water, and wheat. He provided two wooden boxes, round and equal, filled the one with wood, and fufpended an equal weight of gold in the centre of ofcillation 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 refiftance of the air. And placing the one by the other, he observed them to play together for wards and backwards, for a long time, with equal vibrations. And therefore the quantity of matter in the gold was to the quantity of matter in the wood, as the action of the motive force (or VIS MOTRIX) upon all the gold to the action of the fame 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.

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By thefe experiments, in bodies of the fame weight, he could manifeftly have difcovered a difference of matter less than the thousandth part of the whole, had any fuch been. But, without all doubt, the nature of gravity towards the planets, is the fame as towards the earth. fhould 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, fo as to fall together towards the earth; it is certain from what we have demonftrated before, that, in equal times, they would defcribe equal spaces with the moon, and of confequence are to the moon, in quantity of matter as their weights to its weight. Since the fatellites of Jupiter perform their revolutions in times which obferve the fefquiplicate proportion of their diftances from Jupiter's centre, their accelerative gravities towards Jupiter will be reciprocally as the fquares of their diftances from Jupiter's centre; that is, equal at equal distances. And therefore, thefe

fatellites,

moon, to be there compared with its body; if the weights of fuch bodies were to the weights of the external parts of the moon as the quantities of matter in the one and in the other refpc&tively, but to the weights of the internal parts in a greater or lefs proportion; then likewife the weights of thofe bodies would be to the weight of the whole moon in a greater or lefs 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 lefs, according to the variety of forms in equal matter; altogether against experience.

fatellites, if fuppofed to fall towards Jupiter from equal heights, would defcribe equal spaces in equal times, in like manner as heavy bodies do on our earth. And by the fame argument, if the circumfolar planets were fuppofed to be let fall at equal diftances from the fun, they would in their defcent towards the fun, defcribe equal spaces in equal times. But forces, which equally accelerate unequal bodies, must be as thofe bodies: that is to fay, the weights of the planets towards the fun must be as their quantities of matter. Further, that the weights of Jupiter and of his fatellites towards the fun are proportional to the feveral quantities of their matter, appears from the exceeding regular motions of the fatellites. For if fome of thofe bodies were more ftrongly COR. 2. Univerfally, all bodies about the earth attracted to the fun in proportion to their quan- gravitate towards the earth; and the weights of tity of matter than others, the motions of the fa- all, at equal diftances from the earth's centre, are tellites would be difturbed by that inequality of as the quantities of matter which they feverally attraction. If, at equal distances from the fun, contain. This is the quality of all bodies within any fatellite, in proportion to the quantity of its the reach of our experiments; and therefore (by matter, did gravitate towards the fun, with a rule 3.) to be affirmed of all bodies what foever. force greater than Jupiter in proportion to his, If ether, or any other body, were either altogeaccording to any given proportion, fuppofe of d ther void of gravity, or were to gravitate lefs in toe; then the diftance between the centres of the proportion to its quantity of matter; then, befun and of the fatellite's orbit would be always caufe (according to Ariftotle, Des Cartes, and greater than the diftance between the centres of others) there is no difference betwixt that and the fun and of Jupiter nearly in the fubduplicate other bodies, but in mere form of matter, by a of that proportion. And if the fatellite gravitated fucceffive change from form to form, it might be towards the fun with a force lefs in the propor- changed at laft into a body of the fame condition tion of e to d, the diftance of the centre of the with those which gravitate moft in proportion to fatellite's orb from the fun would be less than the their quantity of matter; and, on the other hand, diftance of the centre of Jupiter's from the fun in the heaviest bodies, acquiring the first form of the fubduplicate of the fame proportion. There- that body, might by degrees quite lose their grafore, if at equal distances from the fun, the ac-vity. And therefore the weights would depend celerative gravity of any fatellite towards the fun were greater or lefs than the accelerating gravity of Jupiter towards the fun but by one 1000th. part of the whole gravity; the distance of the centre of the fatellite's orbit from the fun would be greater or less than the distance of Jupiter from the fun by one 200oth part of the whole diftance; that is, by a fifth part of the distance of the ut moft fatellite from the centre of Jupiter; an eccentricity of the orbit which would be very fenfible. But the orbits of the fatellites are concentric to Jupiter; therefore the accelerative gravities of Jupiter, and of all fatellites, towards the fun, are equal among themselves. And by the fame argument, the weight of Saturn and of his fatellites towards the fun, at equal diftances from the fun, are as their feveral quantities of matter; and the weights of the moon and of the earth towards the fun, are either none, or accurately proportional to the maffes of matter which they con

tain.

But further, the weights of all the parts of every planet towards any other planet are one to another as the matter in the several parts. For if fome parts gravitated more, others lefs, than in proportion to the quantity of their matter; then the whole planet, according to the fort of parts with which it moft abounds, would gravitate more or lefs than in proportion to the quantity of matter in the whole. Nor is it of any moment whether thefe parts are external or internal. For if, as an inftance, we should imagine the terreftrial bodies with us to be raised up to the orb of the

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 fpecific gravity of the fluid which fills the region of the air, on account of the extreme denfity of the matter, would fall nothing short of the specific gravity of quick-filver or gold, or any other the most denfe body; and therefore, neither gold, nor any otior body, could defcend in air. For bodies do not defcend 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 diminifhed, what should hinder a diminution to infinity?

COR. 4. If all the folid particles of all bodies are of the fame denfity, nor can be rarefied without pores, a void space or vacuum must be granted. (By bodies of the fame denfity, our author means thofe whofe vires inertia are in the proportion of their bulks.)

PROP. VI. That there is a power of gravity tending to all bodies, proportional to the feveral quantities of matter which they contain. 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, confidered apart, is reciprocally as the square of the distance of places from the centre of the planet. And thence it follows, that the gravity tending towards all the planets is proportional to the matter which they contain. Moreover, fince

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