LEM — 2nd ed. (1778

LEM. III. The same ultimate ratios are also ratios of equality, when the breadths $AB$ , $BC$ , $DC$ , &c. of the parallelograms are unequal, and are all diminished in infinitum. —The demonstration of this differs but little from that of the former.

In his succeeding lemmas, Sir Isaac goes on to prove, in a manner similar to the above, that the ultimate ratios of the sine, chord, and tangent of arcs infinitely diminished, are ratios of equality, and therefore that in all our reasonings about these we may safely use the one for the other: —that the ultimate form of evanescent triangles made by the arc, chord, and tangent, is that of similitude, and their ultimate ratio is that of equality; and hence, in reasonings about

ultimate ratios, we may safely use these triangles for each other, whether made with the sine, the arc, or the tangent. —He then shews some properties of the ordinates of curvilinear figures; and proves that the spaces which a body describes by any finite force urging it, whether that force is determined and immutable, or is continually augmented or continually diminished, are, in the very beginning of the motion, one to the other in the duplicate ratio of the powers. And lastly, having added some demonstrations concerning the evanescence of angles of contact, he proceeds to lay down the mathematical part of his system, and which depends on the following theorems.

THEOR. I. The areas which revolving bodies describe by radii drawn to an immoveable centre of force, lie in the same immoveable planes, and are proportional to the times in which they are described. —For, suppose the time to be divided into equal parts, and in the first part of that time, let the body by its innate force describe the right line $AB$ (fig. 3.); in the second part of that time, the same would, by law 1. if not hindered, proceed directly to $c$ along the line $Bc = AB$ ; so that by the radii $AS$ , $BS$ , $cS$ , drawn to the centre, the equal areas $ASB$ , $BS c$ , would be described. But, when the body is arrived at $B$ , suppose the centripetal force acts at once with a great impulse, and, turning aside the body from the right line $Bc$ , compels it afterwards to continue its motion along the right line $BC$ . Draw $cC$ parallel to $BS$ , meeting $BC$ in $C$ ; and at the end of the second part of the time, the body, by cor. 1. of the laws, will be found in $C$ , in the same plane with the triangle $ASB$ . Join $SC$ ; and because $SB$ and $cC$ are parallel, the triangle $SBC$ will be equal to the triangle $SBC$ , and therefore also to the triangle $SAB$ . By the like argument, if the centripetal force acts successively in $C$ , $D$ , $E$ , &c. and makes the body in each single particle of time to describe the right lines $CD$ , $DE$ , $EF$ , &c. they will all lie in the same 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 composition, any sums $SADS$ , $SAFS$ , of those areas are, one to the other, as the times in which they are described. Now, let the number of those triangles be augmented, and their size diminished 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 described areas $SADS$ , $SAFS$ , which are always proportional to the times of description, will, in this case also, be proportional to those times. Q. E. D.

COR. 1. The velocity of a body attracted towards an immoveable centre, in spaces void of resistance is reciprocally as the perpendicular let fall from that centre on the right line which touches the orbit. For the velocities in these places $A$ , $B$ , $C$ , $D$ , $E$ , are as the bases $AB$ , $BC$ , $DE$ , $EF$ , 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 successively described in equal times by the same body, in spaces void of resistance, are completed into a

parallelogram ABCV, and the diagonal BV of this parallelogram, in the position which it ultimately acquires when those areas are diminished in infinitum is produced both ways, it will pass 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 resistance 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 Bc, BV and Ef, EZ; but BV and EZ, which are equal to Cc and Ef, in the demonstration of this proposition, were generated by the impulses of the centripetal force in B and E, and are therefore proportional to those impulses.

COR. 4. The forces by which bodies, in spaces void of resistance, are drawn back from rectilinear motions, and turned into curvilinear orbits, are one to another as the versed fines of arcs described in equal times; which versed fines tend to the centre of force, and bisect the chords when these arcs are diminished to infinity. For such versed fines are the halves of the diagonals mentioned in cor. 3.

COR. 5. And therefore those forces are to the force of gravity, as the said versed fines to the versed fines perpendicular to the horizon of those parabolic arcs which projectiles describe in the same time.

COR. 6. And the same 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 those planes, are not at rest, but move uniformly forward in right lines.

THEOR. II. Every body that moves in any curve line described in a plane, and, by a radius drawn to a point either immovable or moving forward with an uniform rectilinear motion, describes about that point areas proportional to the times, is urged by a centripetal force directed to that point.

CASE I. For every body that moves in a curve line is (by law 1.) turned aside from its rectilinear course 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 describe in equal times the least equal triangles SAB, SBC, SCD, &c. about the immovable 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 D, that is, in the direction of the line CS, &c; and therefore acts always in the direction of lines tending to the immovable point S. Q. E. D.

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

COR. 1. In non-resisting spaces or 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 consequentia, or towards the parts to which the motion is directed, if the description of the areas is accelerated; but in antecedentia if retarded.

dentia if retarded.

COR. 2. And even in resisting mediums, if the description of the areas is accelerated, the directions of the forces deviate from the point in which the radii meet, towards the parts to which the motion tends.

A body may be urged by a centripetal force compounded of several forces. In which case the meaning of the proposition 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 surface, 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 surface; and is therefore not to be neglected in the composition 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 out of the centripetal forces tending to that other body, and of all the accelerative force by which that other body is impelled.—The demonstration of this is a natural consequence of the theorem immediately preceding.

Hence, if the one body L, by a radius drawn to the other body T, describes areas proportional to the times, and from the whole force by which the first body L is urged, (whether that force is simple, or, according to cor. 2. of the laws, compounded out of several forces), we subduct that whole accelerative force by which the other body is urged; the whole remaining force by which the first body is urged will tend to the other body T, as its centre.

And vice versa, if the remaining force tends nearly to the other body T, those areas will be nearly proportional to the times.

If the body L, by a radius drawn to the other body T, describes areas, which, compared with the times, are very unequal; and that other body T be either at rest, 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 (immovable or moveable) centre. The same thing obtains when the other body is actuated by any other motion whatever; provided that centripetal force is taken which remains after subducting that whole force acting upon that other body T.

Because the equable description of areas indicates that a centre is respected 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 description 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 describe different circles, tend to the centres of the same circles; and are one to the other as the squares of the arcs described in equal times applied to the radii of circles.—For these forces tend

Newtonian Philosophy tend to the centres of the circles, (by theor. 2. and cor. 2. theor. 1.) and are to one another as the versed sines of the least arcs described in equal times, (by cor. 4. theor. 1.) that is, as the squares of the same arcs applied to the diameters of the circles, by one of the lemmas; and therefore, since those arcs are as arcs described in any equal times, and the diameters are as the radii, the forces will be as the squares of any arcs described in the same time, applied to the radii of the circles. Q. E. D.

COR. 1. 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 simple ratio of the radii inversely.

COR. 2. And since the periodic times are in a ratio compounded of the ratio of the radii directly, and the ratio of the velocities inversely; 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 also 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. 5. If the periodic times are as the radii, and therefore the velocities equal, the centripetal forces will be reciprocally as the radii; and the contrary.

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

COR. 7. And universally, if the periodic time is as any power $R^n$ of the radius $R$ , and therefore the velocity reciprocally as the power $R^{n-1}$ of the radius, the centripetal force will be reciprocally as the power $R^{2n-2}$ of the radius; and the contrary.

COR. 8. The same things all hold concerning the times, the velocities, and forces, by which bodies describe the similar parts of any similar figures, that have their centres in a similar position within those figures, as appears by applying the demonstrations of the preceding cases to those. And the application is easy, by only substituting the equable description of areas in the place of equable motion, and using the distances of the bodies from the centres instead of the radii.

COR. 9. From the same demonstration it likewise 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 same body, falling by the same given force, would descend through in the same given time.

“By means of the preceding proposition and its corollaries, (says Sir Isaac), we may discover the proportion of a centripetal force to any other known force, such 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 descent 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 such propositions Mr Huygens, in his excellent book De Horologio Oscillatorio, has compared the force of gravity with the centrifugal forces of revolving bodies.

The preceding proposition may also be demonstrated in the following manner. In any circle suppose a polygon to be inscribed of any number of sides. And if a body, moved with a given velocity along the sides of the polygon, is reflected from the circle at the several angular points; the force with which, at every reflection, it strikes the circle will be as its velocity; and therefore the sum of the forces, in a given time, will be as that velocity and the number of reflections conjointly; that is, (if the species of the polygon be given), as the length described in that given time, and increased or diminished in the ratio of the same length to the radius of the circle; that is, as the square of that length applied to the radius; and therefore, if the polygon, by having its sides diminished in infinitum, coincides with the circle, as the square of the arc described 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 continually repels the body towards the centre, is equal.

On these principles hangs the whole of Sir Isaac Newton's mathematical philosophy. He now shews how to find the centre to which the forces impelling any body are directed, having the velocity of the body given: and finds the centrifugal force to be always as the versed sine of the nascent arc directly, and as the square of the time inversely; or directly as the square of the velocity, and inversely as the chord of the nascent arc. From these premises he deduces the method of finding the centripetal force directed to any given point when the body revolves in a circle; and this whether the central point is near or at an immense distance; so that all the lines drawn from it may be taken for parallels. The same thing he shews with regard to bodies revolving in spirals, ellipses, hyperbolas, or parabolas.—Having the figures of the orbits given, he shews also how to find the velocities and moving powers; and, in short, solves all the most difficult problems relating to the celestial bodies with an astonishing degree of mathematical skill. These problems and demonstrations are all contained in the first book of the Principia: but to give an account of them here would exceed our limits; neither would many of them be intelligible, excepting to first-rate mathematicians.

In the second book Sir Isaac treats of the properties of fluids, and their powers of resistance; and here he lays down such principles as entirely overthrow the doctrine of Des Cartes's vortices, which was the fashionable system in his time. In the third book, he begins particularly to treat of the natural phenomena, and apply to them the mathematical principles formerly demonstrated; and, as a necessary preliminary to this part, he lays down the following rules for reasoning in natural philosophy.

1. We are to admit no more causes of natural things

Newtonian things than such as are both true and sufficient to ex-
Philosophy plain their natural appearances.

2. Therefore to the same natural effects we must always assign, as far as possible, the same causes.

3. The qualities of bodies which admit neither intention nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.

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

The phenomena first considered, are, 1. That the satellites of Jupiter by radii drawn to the centre of their primary, describe areas proportional to the times of their description; and that their periodic times, the fixed stars being at rest, are in the sesquiplicate ratio of their distances from its centre. 2. The same thing is likewise observed of the phenomena of Saturn. 3. The five primary planets, Mercury, Venus, Mars, Jupiter, and Saturn, with their several orbits, encompass the sun. 4. The fixed stars being supposed at rest, the periodic times of the five primary planets, and of the earth, about the sun, are in the sesquiplicate proportion of their mean distances from the sun. 5. The primary planets, by radii drawn to the earth, describe areas no ways proportionable to the times: but the areas which they describe by radii drawn to the sun are proportional to the times of description. 6. The moon, by a radius drawn to the centre of the earth, describes an area proportional to the time of description. All these phenomena are undeniable from astronomical observations, and are explained at large under the article ASTRONOMY. The mathematical demonstrations are next applied by Sir Isaac Newton in the following propositions.

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 squares of the distances of those satellites from that centre. The former part of this proposition appears from theor. 2. or 3. and the latter from cor. 6. of theor. 5.; and the same thing we are to understand of the satellites 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 sun; and are reciprocally as the squares of the distances from the sun's centre. The former part of this proposition is manifest from phenomenon 5. just mentioned, and from theor. 2.; the latter from phenomenon 4. and cor. 6. of theor. 4. But this part of the proposition is with great accuracy deducible from the quiescence of the aphelion points. For a very small aberration from the reciprocal duplicate proportion would produce a motion of the apses, sensible in every single revolution, and in many of them enormously great.

PROP. III. The force by which the moon is retained in its orbit, tends towards the earth; and is reciprocally as the square of the distance of its place from

the centre of the earth. The former part of this proposition is evident from phenomenon 5. and theor. 2.; the latter from phenomenon 6. and theor. 2. or 3. It is also evident from the very slow motion of the moon's apogee; which, in every single revolution, amounting but to $3^{\circ} 3'$ in consequence, may be neglected: and this more fully appears from the next proposition.

PROP. IV. The moon gravitates towards the earth, and by the force of gravity is continually drawn off from a rectilinear motion, and retained in its orbit.—The mean distance of the moon from the earth in the syzgies in semidiameters of the latter, is about $60\frac{1}{2}$ . Let us assume the mean distance of 60 semidiameters in the syzgies; and suppose one revolution of the moon in respect of the fixed stars to be completed in $27^{\circ} 7^{\circ} 43'$ , as astronomers 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 motion, to be let go, so as to descend towards the earth with the impulse of all that force by which it is retained in its orbit, it will, in the space of one minute of time, describe in its fall $15\frac{1}{2}$ Paris feet. For the versed sine of that arc which the moon, in the space of one minute of time, describes by its mean motion at the distance of 60 semidiameters of the earth, is nearly $15\frac{1}{2}$ Paris feet; or more accurately, 15 feet 1 inch and 1 line $\frac{1}{2}$ . Wherefore since that force, in approaching to the earth, increases in the reciprocal duplicate proportion of the distance; and, upon that account, at the surface of the earth is $60 \times 60$ times greater than at the moon; a body in our regions, falling with that force, ought, in the space of one minute of time, to describe $60 \times 60 \times 15\frac{1}{2}$ Paris feet; and in the space of one second of time to describe $15\frac{1}{2}$ of those feet; or, more accurately, 15 feet 1 inch, 1 line $\frac{1}{2}$ . And with this very force we actually find that bodies here on earth do really descend. For a pendulum oscillating seconds in the latitude of Paris, will be three Paris feet and $8\frac{1}{2}$ lines in length, as Mr Huygens has observed. And the space which a heavy body describes by falling one second 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 $\frac{1}{2}$ . And therefore the force by which the moon is retained in its orbit, becomes, at the very surface of the earth, equal to the force of gravity which we observe in heavy bodies there. And therefore (by rule 1. and 2.) the force by which the moon is retained in its orbit is that very same force which we commonly call gravity. For were gravity another force different from that, then bodies descending to the earth with the joint impulse of both forces, would fall with a double velocity, and, in the space of one second of time, would describe 30 $\frac{1}{2}$ Paris feet; altogether against experience.

The demonstration of this proposition may be more diffusely explained after the following manner. Suppose several moons to revolve about the earth, as in the system of Jupiter or Saturn; the periodic times of those moons would (by the argument of induction) observe the same law which Kepler found to obtain among the planets; and therefore their centripetal forces would be reciprocally as the squares of the distances from the centre of the earth by Prop. I. Now, if the

Newtonian Philosophy the lowest of these were very small, and were so near the earth as almost to touch the tops of the highest mountains, the centripetal force thereof, retaining it in its orbit, would be very nearly equal to the weights of any terrestrial bodies that should be found upon the tops of these mountains; as may be known from the foregoing calculation. Therefore if the same little moon should be deserted by its centrifugal force that carries it through its orb, it would descend to the earth; and that with the same velocity as heavy bodies do actually descend with upon the tops of those very mountains, because of the equality of forces that oblige them both to descend. And if the force by which that lowest moon would descend were different from that of gravity, and if that moon were to gravitate towards the earth, as we find terrestrial bodies do on the tops of mountains, it would then descend with twice the velocity, as being impelled by both these forces conspiring together. Therefore, since both these forces, that is, the gravity of heavy bodies, and the centripetal forces of the moons, respect the centre of the earth, and are similar and equal between themselves, they will (by rule 1. and 2.) have the same cause. And therefore the force which retains the moon in its orbit, is that very force which we commonly call gravity; because otherwise this little moon at the top of a mountain must either be without gravity, or fall twice as swiftly as heavy bodies use to do.

Having thus demonstrated that the moon is retained in its orbit by its gravitation towards the earth, it is easy to apply the same demonstration to the motions of the other secondary planets, and of the primary planets round the sun, and thus to shew that gravitation prevails throughout the whole creation; after which, Sir Isaac proceeds to shew from the same principles, that the heavenly bodies gravitate towards each other, and contain different quantities of matter, or have different densities in proportion to their bulks.

PROP. V. All bodies gravitate towards every planet; and the weights of bodies towards the same 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 sorts of heavy bodies (allowance being made for the inequality of retardation by some small resistance of 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. Sir Isaac Newton tried the thing in gold, silver, lead, glass, sand, common salt, wood, water, and wheat. He provided two wooden boxes, round and equal, filled the one with wood, and suspended an equal weight of gold 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, he observed them to play together forwards 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 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 experi-

ments, in bodies of the same weight, he could manifestly 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 demonstrated 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 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 exceeding regular motions of the satellites. 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 satellites would be disturbed by that inequality of attraction. If, at equal distances from the sun, any satellite, in proportion 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 proportion, suppose of $d$ to $e$ ; then the distance between the centres of the sun and of the satellite's orbit would be always greater than the distance between the centres of the sun and of Jupiter, nearly in the subduplicate of that proportion. And if the satellite gravitated towards the sun with a force, lesser in the proportion of $d$ to $e$ , the distance of the centre of the satellite's orb from the sun, would be less than the distance of the centre of Jupiter's from the sun, in the subduplicate 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 accelerating gravity of Jupiter towards the sun, but by one $\frac{1}{1000}$ 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 $\frac{1}{1000}$ part of the whole distance; that is, by a fifth 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; therefore the accelerative gravities of Jupiter, and of all its satellites, towards the sun, are equal among themselves. And by the same argument, the weight of Saturn and of his satellites towards the sun, at equal distances from the sun, are as their several quantities of

Newtonian of matter; and the weights of the moon and of the
Philosophy earth towards the sun, are either none, or accurately
proportional to the masses 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 some parts gra-
vitated 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 external 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 de-
pend 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 gra-
vitate towards the earth; and the weights of all, at
equal distances from the earth's centre, are as the quan-
tities of matter which they severally contain. This is
the quality of all bodies within the reach of our experi-
ments; and therefore (by rule 3.) to be affirmed of
all bodies whatsoever. If the ether, or any other body,
were either altogether void of gravity, or were to gra-
vitate less in proportion to its quantity of matter;
then, because (according to Aristotle, 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 mat-
ter; and, on the other hand, the heaviest bodies, ac-
quiring 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 quick-silver, or gold,
or any other the most dense body; and therefore, nei-
ther 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 quan-
tity of matter in a given space can by any rarefaction
be diminished, what should hinder a diminution to in-
finity?

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. [By bodies of
the same density, our author means those whose vires

inertie are in the proportion of their bulks.]

PROP. VI. That there is a power of gravity tend-
ing to all bodies, proportional to the several 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, considered
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 pla-
nets, is proportional to the matter which they contain.

Moreover, since all the parts of any planet A gra-
vitate 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 3.)
to every action corresponds an equal re-action: there-
fore the planet B will, on the other hand, gravitate to-
wards 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
attractions towards the whole arises from the attractions
towards the several parts. The thing may be easily un-
derstood in gravity, if we consider a greater planet as
formed of a number of lesser planets, meeting together
in one globe. For hence it would appear that the
force of the whole must arise from the forces of the
component parts. If it be objected, That, according
to this law, all bodies with us must mutually gravitate
one towards another, whereas no such gravitation any-
where appears; it is answered, That since the gravita-
tion towards these bodies is to the gravitation towards
the whole earth, as these bodies are to the whole earth,
the gravitation towards them must be far less than to
fall under the observation of our senses. [The experi-
ments with regard to the attraction of mountains,
however, have now further elucidated this point.]

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 particles.

PROP. VII. In two spheres mutually gravitating
each towards the other, if the matter, in places on all
sides round about and equidistant from the centres, is
similar; the weight of either sphere towards the other,
will be reciprocally as the square of the distance be-
tween their centres.

For the demonstration of this, see the Principia,
book i. prop. 75 and 76.

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 pla-
nets, are 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 Ve-
nus, revolving about the sun, in 224^d. 16^h 4^m; of the ut-
most circumjovial satellite revolving about Jupiter, in
16^d. 16^h 4^m; of the Huygenian satellite about Saturn
in 15^d. 22^h 3^m; and of the moon about the earth in 27^d.

Newtonian 7". 43'; compared with the mean distance of Venus
Philosophy 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 computation our author 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 $10000:997:791:109$ respectively. Then, because, as the distances are increased or diminished, 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.

COR. 2. Hence likewise we discover the quantity of matter in the several 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 $1:10000:997:791$ 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 proportion.

COR. III. Hence also we find the densities of the planets. For (by prop. 72. book 1.) the weights of equal and similar bodies towards similar spheres, are, at the surfaces of those spheres, as the diameters of the spheres. 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, Jupiter, 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 respectively; and therefore their densities are as 100, 94 $\frac{1}{2}$ , 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.

COR. 4. The smaller the planets are, they are, ceteris paribus, of so much the greater density. For so the powers of gravity on their several surfaces come nearer to equality. They are likewise, ceteris paribus, 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 of the sun's heat. Our water, if it were removed as far as the orb of Saturn, would be converted into ice, and in the orb of Mercury would quickly fly away in vapour. 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 Sir Isaac 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. Niagara.

It is shewn in the scholium of prop. 22. book 2. of the Principia, that at the height of 200 miles above the earth, the air is more rare than it is at the superficies of the earth, in the ratio of 30 to 0.00000000000003998, or as 750000000000 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 100000th 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 therefore the celestial regions being perfectly void of air and exhalations, the planets and comets meeting no sensible resistance in those spaces, will continue their motions through them for an immense space of time.