of all bodies whatsoever. That all bodies are rnoveable, and endowed with
certain powers (which we call the vires inertias] of persevering in their mo
tion, or in their rest, we only infer from the like properties observed in the
BOOK 1II.J OF NATURAL PHILOSOPHY. 385
bodies which we have seen. The extension, hardness, impenetrability, mo
bility, and vis inertia of the whole, result from the extension, hardness,
impenetrability, mobility, and vires inertia of the parts; and thence we
conclude the least particles of all bodies to be also all extended, and hard
and impenetrable, and moveable, and endowed with their proper vires inertia.
And this is the foundation of all philosophy. Moreover, that the divided
but contiguous particles of bodies may be separated from one another, is
matter of observation ; and, in the particles that remain undivided, our
minds are able to distinguish yet lesser parts, as is mathematically demon
strated. But whether the parts so distinguished, and not yet divided, may,
by the powers of Nature, be actually divided and separated from one an
other, we cannot certainly determine. Yet, had we the proof of but one
experiment that any undivided particle, in breaking a hard and solid body,
suffered a division, we might by virtue of this rule conclude that the un
divided as well as the divided particles may be divided and actually sep
arated to infinity.
Lastly, if it universally appears, by experiments and astronomical obser
vations, that all bodies about the earth gravitate towards the earth, and
that in proportion to the quantity of matter which they severally contain ;
that the moon likewise, according to the quantity of its matter, gravitates
towards the earth
; that, on the other hand, our sea gravitates towards the
moon ; and all the planets mutually one towards another ; and the comets
in like manner towards the sun ; we must, in consequence of this rule, uni
versally allow that all bodies whatsoever are endowed with a principle ot
mutual gravitation. For the argument from the appearances concludes with
more force for the universal gravitation of all bodies than for their impen
etrability ; of which, among those in the celestial regions, we have no ex
periments, nor any manner of observation. Not that I affirm gravity to be
essential to bodies : by their vis insita I mean nothing but their vis iiicrticz.
This is immutable. Their gravity is diminished as they recede from the
earth.
RULE IV.
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.
This rule we must follow, that the argument of induction may not bf
evaded by hypotheses.
25
386 THE MATHEMATICAL PRINCIPLES [BooK III.
PHENOMENA, OR APPEARANCES,
PHENOMENON I.
That the circumjovial planets, by radii drawn to Jupiter s centre, de
scribe areas proportional to the times of description ; and that their
periodic times, the fixed stars being at rest, are in the sesquiplicate
proportion of their distances from, its centre.
This we know from astronomical observations. For the orbits of these
planets differ but insensibly from circles concentric to Jupiter ; and their
motions in those circles are found to be uniform. And all astronomers
agree that their periodic times are in the sesquiplicate proportion of the
semi-diameters of their orbits; and so it manifestly appears from the fol-
1owing table.
The periodic times of the satellites of Jupiter. H 18h
. 27 . 34". 3d
. 13h
. 13 42". 7d
. 31
. 42 36". 16d
. 16h
. 32 9".
The distances of the satellites from Jupiter s centre.
Mr. Pound has determined, by the help of excellent micrometers, the
diameters of Jupiter and the elongation of its satellites after the following
manner. The greatest heliocentric elongation of the fourth satellite from
Tupiter s centre was taken with a micrometer in a 15 feet telescope, and at
the mean distance of Jupiter from the earth was found about 8 16". The
elongation of the third satellite was taken with a micrometer in a telescope
of 123 feet, and at the same distance of Jupiter from the earth was found
4 42". The greatest elongations of the other satellites, at the same dis
tance of Jupiter from the earth, are found from the periodic times to be 2
56" 47 ", and 1 51" 6 ".
The diameter of Jupiter taken with the micrometer in a 123 feet tele
scope several times, and reduced to Jupiter s mean distance from the earth,
proved always less than 40", never less than 38", generally 39". This di
ameter in shorter telescopes is 40", or 41"; for Jupiter s light is a little
dilated by the unequal refrangibility of the rays, and this dilatation bears
3 less ratio to the diameter of Jupiter in the longer and more perfect teleescopes
than in those which are shorter and less perfect. The times :i
HOOK. III.] OF NATURAL PHILOSOPHY 387
which two satellites, the first and the third, passed over Jupiter s body, were
observed, from the beginning of the ingress to the beginning of the egress,
and from the complete ingress to the complete egress, with the long tele
scope. And from the transit of the first satellite, the diameter of Jupiter
at its mean distance from the earth came forth 37 J-". and from the transit
of the third 371". There was observed also the time in which the shadow
of the first satellite passed over Jupiter s body, and thence the diameter of
Jupiter at its mean distance from the earth came out about 37". Let us
suppose its diameter to be 37}" very nearly, and then the greatest elonga
tions of the first, second, third, and fourth satellite will be respectively
equal to 5,965, 9,494, 15,141, and 26,63 semi-diameters of Jupiter.
PHENOMENON II.
Tkat the. circumsalurnal planets, by radii drawn, to Saturtfs centre, de
scribe areas proportional to the times of description ; and that their
periodic times, the fixed stars being at rest, are in the sesqniplicata
proportion uf their distances from its centre.
For, as Cassiui from his own observations has determined, theii distan
ces from Saturn s centre and their periodic times are as follow.
The periodic times of the satellites of Saturn.
l d
. 2l h
. IS 27". 2d
. 17h
. 41 22". 4d
. 12". 25 12". 15d
. 22^. 41 14",
79 1
. 71
. 48 00".
The distances of the satellitesfrom Saturn s centre, in semi-diameters oj
itv ring.
From observations li-. 2f 3|. 8. 24
From the periodic times . . . 1,93. 2,47. 3,45. 8. 23.35.
The greatest elongation of the fourth satellite from Saturn s centre is
commonly determined from the observations to be eight of th-se semidiameters
very nearly. But the greatest elongation of this satellite from
Saturn s centre, when taken with an excellent micrometer iuMr../fuygen8>
telescope of 123 feet, appeared to be eight semi-diameters and T
7- of a semidiameter.
And from this observation arid the periodic times the distances
of the satellites from Saturn s centre in serni-diameters of the ring are 2.1.
2,69. 3,75. 8,7. and 25,35. The diameter of Saturn observed in the same
telescope was found to be to the diameter of the ring as 3 to 7 ; and the
diameter of the ring, May 28-29, 1719, was found to be 43"
; and th:*nce
the diameter of the ring when Saturn is at its mean distance from the
earth is 42", and the diameter of Saturn 18". These things appear so in
very long and excellent telescopes, because in such telescopes the apparent
magnitudes of the heavenly bodies bear a greater proportion to the dilata
tion of light in the extremities of those bodies than in shorter telescopes.
3S8 THE MATHEMATICAL PRINCIPLES [HoOK III
If we, then, reject all the spurious light, the diameter of Saturn will not
amount to more than 16".
PHENOMENON III.
That the five primary planets, Mercury, Venus, Mars, Jupiter, and Sat
urn, with their several orbits, encompass the sun.
That Mercury and Venus revolve about the sun, is evident from their
moon-like appearances. When they shine out with a full face, they are, in
respect of us, beyond or above the sun ; when they appear half full, they
are about the same height on one side or other of the sun ; when horned,
they are below or between us and the sun ; and they are sometimes, when
directly under, seen like spots traversing the sun s disk. That Mars sur
rounds the sun, is as plain from its full face when near its conjunction with
the sun. and from the gibbous figure which it shews in its quadratures.
And the same thing is demonstrable of Jupiter and Saturn, from their ap
pearing full in all situations ;
for the shadows of their satellites that appear
sometimes upon their disks make it plain that the light they shine with is
not their own, but borrowed from the sun.
PHENOMENON IV.
That the fixed stars being at rest, the periodic times of the five primary
planets, and (whether of the suit about the earth, or) of the earth about
the sun, are in the sesquiplicate proportion of their mean distances
from the sun.
This proportion, first observed by Kepler, is now received by all astron
omers ; for the periodic times are the same, and the dimensions of the orbits
are the same, whether the sun revolves about the earth, or the earth about
the sun. And as to the measures of the periodic times, all astronomers are
agreed about them. But for the dimensions of the orbits, Kepler and Bullialdns,
above all others, have determined them from observations with the
greatest accuracy ; and the mean distances corresponding to the periodic
times differ but insensibly from those which they have assigned, and for
the most part fall in between them ; as we may see from the following table.
The periodic times with respect to the fixed stars, of the planets and earth
revolving about the sun. in days and decimal parts of a day.
* ^ * $ ? *
10759,275. 4332,514. 686,9785. 365,2565. 224,6176. 87,9692.
The mean distances of the planets and of the earth from the sun.
* V I
According to Kepler 951000. 519650. 152350.
to Bullialdus 954198. 522520. 152350.
to the periodic times .... 954006. 520096. 152369
BOOK III.] OF NATURAL PHILOSOPHY. 389
J ? *
According to Kepler 100000. 72400. 38806
" to Bnllialdus ... . . . 100000. 72398. 38585
" to the periodic times 100000. 72333. 38710.
As to Mercury and Venus, there can be no doubt about their distances
from the sun ;
for they are determined by the elongations of those planets
from the sun ; and for the distances of the superior planets, all dispute is
cut off by the eclipses of the satellites of Jupiter. For by those eclipses
the position of the shadow which Jupiter projects is determined ; whence
we have the heliocentric longitude of Jupiter. And from its helio
centric and geocentric longitudes compared together, we determine its
distance.
PHENOMENON V.
Then the primary planets, by radii drawn to the earth, describe areas no
wise proportional to the times ; but that the areas which they describe
by radii drawn to the snn are proportional to the times of descrip
tion.
For to the earth they appear sometimes direct, sometimes stationary,
nay, and sometimes retrograde. But from the sun they are always seen
direct, and to proceed with a motion nearly uniform, that is to say, a little
swifter in the perihelion and a little slower in the aphelion distances, so as
to maintain an equality in the description of the areas. This a noted
proposition among astronomers, and particularly demonstrable in Jupiter,
from the eclipses of his satellites; by the help of which eclipses, as we have
said, the heliocentric longitudes of that planet, and its distances from the
sun, are determined.
PHENOMENON VI.
That the moon, by a radius drawn to the earths centre, describes an area
proportional to the time of description.
This we gather from the apparent motion of the moon, compared with
its apparent diameter. It is true that the motion of the moon is a little
disturbed by the action of the sun : but in laying down these Phenomena
I neglect those imall and inconsiderable errors.
390 THE MATHEMATICAL PRINCIPLES [BOOK III
PROPOSITIONSPROPOSITION
I. THEOREM I.
That the forces by which the circumjovial planets are continually drawn
offfrom rectilinear motions, and retained in their proper orbits, tend
to Jupiter s centre ; and are reciprocally as the squares of the distances
of the places of those planets/ro?/i that centre.
The former part of this Proposition appears from Pham. I, and Prop.
II or III, Book I : the latter from Phaen. I, and Cor. 6, Prop. IV, of the same
Book.
The same thing we are to understand of the planets which encompass
Saturn, by Phaon. II.
PROPOSITION II. THEOREM II.
That 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 of the
places of those planets from the sun s centre.
The former part of the Proposition is manifest from Phasn. V, and
Prop. II, Book I
; the latter from Phaen. IV, and Cor. 6, Prop. IV, of the
same Book. But this part of the Proposition is, with great accuracy, de
monstrable from the quiescence of the aphelion points ; for a very small
aberration from the reciprocal duplicate proportion would (by Cor. 1, Prop.
XLV, Book I) produce a motion of the apsides sensible enough in every
single revolution, and in many of them enormously great.
PROPOSITION III. THEOREM III.
That the force by which the moon is retained in its orbit tends to the
earth ; and is reciprocally as the square of the distance of its
plac>>,
from the earths centre.
The former part of the Proposition is evident from Pha3n. VI, and Prop.
II or III, Book I
; the latter from the very slow motion of the moon s apo
gee; which in every single revolution amounting but to 3 3 in consequentia,
may be neglected. For (by Cor. 1. Prop. XLV, Book I) it ap
pears, that, if the distance of the moon from the earth s centre is to the
semi-diameter of the earth as D to 1, the force, from which such a motion
will result, is reciprocally as D 2
^f 3, i. e., reciprocally as the power of D,
whose exponent is 2^^ ; that is to say, in the proportion of the distance
something greater than reciprocally duplicate, but which comes 59f time?
nearer to the duplicate than to the triplicate proportion. But in regard
that this motion is owinsr to the action of the sun (as we shall afterwards
BOOK III.] OF NATURAL PHILOSOPHY. 391
shew), it is here to be neglected. The action of the sun, attracting the
moon from the earth, is nearly as the moon s distance from the earth ; and
therefore (by what we have shewed in Cor. 2, Prop. XLV. Book I) is to the
centripetal force of the moon as 2 to 357,45, or nearly so
; that is, as 1 to
178 f-
. And if we neglect so inconsiderable a force of the sun, the re
maining force, by which the moon is retained in its orb, will be recipro
cally as D 2
. This will yet more fully appear from comparing this force
with the force of gravity, as is done in the next Proposition.
COR. If we augment the mean centripetal force by which the moon is
retained in its orb, first in the proportion of 177%$ to 178ff, and then in
the duplicate proportion of the semi-diameter of the earth to the mean dis
tance of the centres of the moon and earth, we shall have the centripetal
force of the moon at the surface of the earth ; supposing this force, in de
scending to the earth s surface, continually to increase in the reciprocal
duplicate proportion of the height.
PROPOSITION IV. THEOREM IV.
That the moon gravitates towards the earth, and by thejorce oj 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 syzygies in semidiameters
of the earth, is, according to Ptolemy and most astronomers,
59 : according to Vendelin and Huygens, 60 ;
to Copernicus, 60 1 ;
to
Street, 60| ; and to Tycho, 56|. But Tycho, and all that follow his ta
bles of refraction, making the refractions of the sun and moon (altogether
against the nature of light) to exceed the refractions of the fixed stars, and
that by four or five minutes near the horizon, did thereby increase the
moon s horizontal parallax by a like number of minutes, that is, by a
twelfth or fifteenth part of the whole parallax. Correct this error, and
the distance will become about 60^ semi-diameters of the earth, near to
what others have assigned. Let us assume the mean distance of 60 diam
eters in the syzygies ; and suppose one revolution of the moon, in respect
of the fixed stars, to be completed in 27d
. 7h
. 43 , as astronomers have de
termined ; and the circumference of the earth to amount to 123249600
Paris feet, as the French have found by mensuration. And 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 (by Cor.
Prop. Ill) it is retained in its orb, it will in the space of one minute of time,
describe in its fall 15 T^ Paris feet. This we gather by a calculus, founded
either upon Prop. XXXVI, Book [, or (which comes to the same thing;
upon Cor. 9, Prop. IV, of the same Book. For the versed sine of that arc,
which the moon, in the space of one minute of time, would by its mean
392 THE MATHEMATICAL PRINCIPLES [BOOK III
motion describe at the distance of 60 seini-diameters of the earth, is nearly
15^ Paris feet, or more accurately 15 feet, 1 inch, and 1 line . Where
fore, since that force, in approaching to the earth, increases in the recipro
cal duplicate proportion of the distance, and, upon that account, at the
surface of the earth, is 60 X 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 X 60 X 15 T
] Paris feet; and, in the space of one sec
ond of time, to describe 15 ,\ of those feet; or more accurately 15 feet, 1
inch, and 1 line f. And with this very force we actually find that bodies
here upon earth do really descend : for a pendulum oscillating seconds in
the latitude of Paris will be 3 Paris feet, and 8 lines 1 in length, as Mr.
Hu.y veus has observed. And the space which a heavy body describes
by falling in one second of time is to half the length of this pendulum in
the duplicate ratio of the circumference of a circie to its diameter (as Mr.
Htiy^ens has also shewn), and is therefore 15 Paris feet, I inch, 1 line J.
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 ob
serve in heavy bodies there. And therefore (by Rule I and II) 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^ Paris feet ; altogether against experience.
This calculus is founded on the hypothesis of the earth s standing still
;
for if both earth and moon move about the sun. and at the same time about
their common centre of gravity, the distance of the centres of the moon and
earth from one another will be 6(H semi-diameters of the earth ;
as may
be found by a computation from Prop. LX, Book I.
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 reciprocally as the squares of the distances from the centre
of the earth, by Prop. I, of this Book. Now if the lowest of these were
very small, and were so near the earth as almost to touo the tops of the
highest mountains, the centripetal force thereof, retaining it in its orb,
would be very 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 orb, and so be
BOOK 111.] OF NATURAL PHILOSOPHY. 393
lisabled from going onward therein, it would descend to the earth ; and
that with the same velocity as heavy bodies do actually fall with upo-n 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 that lowest moon
would descend were different from gravity, and if that moon were to gravi
tate towards the earth, as we find terrestrial bodies do upon the tops of
mountains, it would then descend with twice the velocity, as being impel
led 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 I and II) 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. THEOREM V.
Vhat the circumjovial planets gravitate towards Jupiter ; the circnntsaturnal
towards Saturn ; the circumsolar towards the sun ; and by t/ie
forces of their gravity are drawn off from rectilinear motions, and re
tained in curvilinear orbits.
For the revolutions of the circumjovial planets about Jupiter, of the
circumsaturnal about Saturn, and of Mercury and Venus, and the other
circumsolar planets, about the sun, are appearances of the same sort with
the revolution of the moon about the earth
; and therefore, by Rule II,
must be owing to the same sort of causes ; especially since it has been
demonstrated, that the forces upon which those revolutions depend tend to
the centres of Jupiter, of Saturn, and of the sun ; and that those forces, in
receding from Jupiter, from Saturn, and from the sun, decrease in the same
proportion, and according to the same law, as the force of gravity does in
receding from the earth.
COR. 1. There is, therefore, a power of gravity tending to all the plan
ets ; for, doubtless, Venus, Mercury, and the rest, are bodies of the same
sort with Jupiter and Saturn. And since all attraction (by Law III) is
mutual, Jupiter will therefore gravitate towards all his own satellites, Sat
urn towards his, the earth towards the moon, and the sun towards all the
primary planets.
COR. 2. The force of gravity which tends to any one planet is re
ciprocally as the square of the distance of places from that planet s
centre.
