COR. 3. All the planets do mutually gravitate towards one another, by
Cor. 1 and 2. And hence it is that Jupiter and Saturn, when near their
394 THE MATHEMATICAL PRINCIPLES [BOOK III
conjunction; by their mutual attractions sensibly disturb each other s ?n>
tions. So the sun disturbs the motions of the moon ; and both sun ini
moon disturb our sea, as we shall hereafter explain.
SCHOLIUM.
The force which retains the celestial bodi in their orbits has been
hitherto called centripetal force; but it being now made plain that it can
be no other than a gravitating force, we shall hereafter call it gravity.
For the cause of that centripetal force which retains the moon in its orbit
will extend itself to all the planets, by Rule I, II, and IV.
PROPOSITION VI. THEOREM VI.
That all bodies gravitate towards every planet ; and that the weights of
bodies towards any the same planet, at equal distances from the centre
of the planet, are proportional to the quantities of matter which they
severally contain.
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, eommpn 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, wi h equal vibrations. And therefore the quantity of matte* : n the
gold (by Cor. 1 and 6, Prop. XXIV, Book II) was to the quantity ot mat
ter in the wood as the action of the motive force (or vis tnotrix) 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, 1 could man
ifestly have discovered a difference of matter less than the thousandth part
of the whol^, 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
HOOK ill.] or NVTURAL PHILOSOPHY, 395
their revolutions in times which observe the sesquiphiate pr portion ol
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 sa_y, the weights
;f 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. LXV, Book
1). 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.^, 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 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 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 d, 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 T oV 7
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 ^oVo Part of the whole distance; that is, by a nf h 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. But some they are, by Cor. 1 and 3, Prop. V.
But further ; the weights of all the parts of every planet f awards any other
396 THE MATHEMATICAL PRINCIPLES [BOOK II]
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 ather, or anj
other body, were either altogether void of gravity, or were to gravitate lesr
in proportion to its quantity of matter, then, because (according to Aris
totle, Des Carles, 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 -acuum must be granted
BOOK Ill.J OF NATURAL PHILOSOPHY. 397
By bodies of the same density, I mean those whose vires inertia 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 VII. THEOREM VII.
That there is a power of gravity tending 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 (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 would 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. LXXIV, Book I.
39S THE MATHEMATICAL PRINCIPLES [BOOK III
PROPOSITION VIII. THEOREM VIII.
Tn two spheres mutually gravitating each towards the other, if tlie matter
in places on all sides round about and equi-distantfrom the centres is
similar, the weight of either sphere towards the other will be recipro
cally as the square of the distance between their centres.
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 was 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<J
. 16f
h
, of the utmost circumjovial satellite
revolving about Jupiter, in 16 . 10 -?/. ; of the Huygenian satellite about
Saturn in 15d
. 22f
h
.
; and of the moon about the earth in 27d
. 7h
. 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"
;
arid 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, T ^VT> ^oVr? an^
___i___
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 II1.J OF NATURAL PHILOSOPHY. 39(
.*
planets; for their quantities of matter are as the forces of gravity at equai
distances from their centres; that is, in the sun, Jupiter, Saturn, and the
earth, as 1, TO FTJ a-oVr? anc^ TeVaja 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 5
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, arid
109; and the weights towards the same as 10000, 943, 529, and 435 re
spectively ; and therefore their densities are as 100. 94|, 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, cccteris parilms, of so
much the greater density ;
for so the powers of gravity on their several
surfaces come nearer to equality. They are likewise, cccteris paribiis, 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 downward from t/ie surface
of the planets decreases nearly 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,
100 THE MATHEMATICAL PRINCIPLES [BOOK III
therefore, can be no greater than what may arise from the inequality of
the density.
PROPOSITION X. THEOREM X.
That the 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 3\6 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 consisted of
water only, whatever was of less density than water, because of its Ivss
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 matter 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. If,
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 Avith 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 0,0000000000003999, 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 t 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 THEOREM XI.
That the common, centre of gravity of the earth, the sun, and all the
planets, is immovable.
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 XII. THEOREM XII.
That the sun is agitated by a perpetual motion, but never recedes jar
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
102 THE MATHEMATICAL PRINCIPLES [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.
Con. 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
