BOOK THREE
ON GRAVITY
PROPOSITION IV
That the moon gravitates toward the earth, and by the force of gravity
is continually drawn off from a rectilinear motion and retained in its
orbit.
SCHOLIUM
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 these moons
(by the argument of induction) would observe the same law which Kepler
found to obtain among the planets, and therefore their centripetal forces
would be inversely as the squares of the distances from the center of the
earth, by Prop. 1 of this book. Now if 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 in its orbit, would
be nearly equal to the weights of any terrestrial bodies that should be
found upon the tops of those mountains, as may be known by the foregoing
computation. Therefore, if the same little moon should be deserted by
its centrifugal force that carries it through its orbit, and be disabled
from going onward therein, it would descend to the earth; and that with
the same velocity with which heavy bodies actually fall upon the tops of
those very mountains, because of the equality of the forces that oblige
them both to descend. And if the force by which the lowest moon would
descend were different from gravity, and if the moon were to gravitate
toward the earth, as we find terrestrial bodies do upon 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, are directed to the center of the earth and are similar and equal
between themselves, they will (by Rules 1 and 2) have one and 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 are wont to do.
PROPOSITION V
That the circumjovial planets gravitate towards Jupiter; the circimsaturnal
towards Saturn; the circumsolar towards the sun; and by the forces of their
gravity are drawn off from rectilinear motions, and retained in curvilinear
orbits.
SCHOLIUM
The force which retains the celestial bodies 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 Rules 1, 2, and 4.
PROPOSITION VI
That all bodies gravitate toward every planet; and that the weights
of bodies toward any one planet, at equal distances from the center of
the planet, are proportional to the quantities of matter which they severally
contain.
It has been now for 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
experiments with gold, silver, lead, glass, sand, common salt, wood, water,
and wheat. I provided two wooden boxes, round and equal: I filled the
one with wood, and suspended an equal weight of gold (as exactly as I could)
in the center of oscillation of the other. The boxes, hanging by equal
threads of 11 feet, made a couple of pendulums perfectly equal in weight
and figure and equally receiving the resistance of the air. And, placing
the one by the other, I observed them to play together forward and backward
for a long time with equal vibrations. And therefore the quantity of
matter in the gold (by Cor. I and VI, Prop. 24, Book II) 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 experiments, in bodies of the same weight,
I 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 toward the planets is the same as toward the earth.
For, should we imagine our terrestrial bodies taken to the orbit of the
moon and there, together with the moon, deprived of all motion, to be let
go, so as to fall together toward 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 3/2th power
of the proportion of their distances from Jupiter’s center, their accelerative
gravities toward Jupiter will be inversely as the squares of their distances
from Jupiter’s center; that is, equal, at equal distances. And therefore
these satellites, if supposed to fall toward 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 toward 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 toward the
sun must be as their quantities of matter. Further, that the weights
of Jupiter and of his satellites toward the sun are proportional to the
several quantities of their matter, appears from the exceedingly regular
motions of the satellites (by Cor. III, Prop. 65, 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 satellites would
be disturbed by that inequality of attraction (by Cor. II, Prop. 65, Book
1). If, at equal distances from the sun, any satellite, in proportion
to the quantity of its matter, did gravitate toward 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 centers
of the sun and of the satellite’s orbit would be always greater than the
distance between the centers of the sun and of Jupiter, nearly as the square
root of that proportion: as by some computations I have found. And if
the satellite did gravitate toward the sun with a force less in the proportion
of e to
d, the distance of the center of the satellite’s
orbit from the sun would be less than the distance of the center of Jupiter
from the sun as the square root of the same proportion. Therefore if,
at equal distances from the sun, the accelerative gravity of any satellite
toward the sun were greater or less than the accelerative gravity of Jupiter
toward the sun but by one 1/1,000 part of the whole gravity, the distance
of the center of the satellite’s orbit from the sun would be greater or
less than the distance of Jupiter from the sun by one 1/2,000 part of the
whole distance; that is, by a fifth part of the distance of the utmost
satellite from the center 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 toward the sun are equal among themselves. And by
the same argument the weights of Saturn and of his satellites toward 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 toward the sun are
either none, or accurately proportional to the masses of matter which they
contain. But some weight they have, by Cor. I and III, Proposition 5.
But further; the weights of all the parts of every planet toward any
other 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 to the orbit 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 shown above.
COR. I. 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. II. Universally, all bodies about the
earth gravitate toward the earth; and the weights of all, at equal distances
from the earth’s center, 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 3) to be affirmed of all bodies whatsoever. If
the ether, or any other body, were either altogether void of gravity or
were to gravitate less in proportion to its quantity of matter, then, because
(according to Aristotle, Descartes, and others) there is no difference
between 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. III. 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. IV. If all the solid particles of all
bodies are of the same density and cannot be rarefied without pores, then
a void space or vacuum must be granted. By bodies of the same density
I mean those whose inertias are in the proportion of their bulks.
COR. V. 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 as the square, but almost as the cube of
the distance, as nearly as I could judge from some rude observations.
[End of Book 3. Rules of Reasoning in Philosophy. On
Gravity]

