to the centre of the earth are (by Prop. 1, Book 1, Princip. Math.} propor
tional to the times in which they are described, its velocity, when it returns
*o the mountain, will be no less than it was at first; and, retaining the
*ame velocity, it will describe the same curve over and over, by the same law
514 THE SYSTEM OF THE WORLD.
But if we now imagine bodies to be projected in the directions of lines
parallel to the horizon from greater heights, as of 5, 10, 100, 1000, or more
miles, or rather as many semi-diameters of the earth, those bodies, accord
ing to their different velocity, and the different force of gravity in different
heights, will describe arcs either concentric with the earth, or variously
eccentric, and go on revolving through the heavens in those trajectories,
just as the planets do in their orbs.
As when a stone is projected obliquely, that is, any way but in the per
pendicular direction, the perpetual deflection thereof towards the earth
from the right line in which it was projected is a proof of its gravitation
to the earth, no less certain than its direct descent when only suffered to
fall freely from rest ; so the deviation of bodies moving in free spaces from
rectilinear paths, and perpetual deflection therefrom towards any place, is
a sure indication of the existence of some force which from all quarters
impels those bodies towards that place.
And as, from the supposed existence of gravity, it necessarily follows
that all bodies about the earth must press downwards, and therefore must
either descend directly to the earth, if they are let fall from rest, or at
least perpetually deviate from right lines towards the earth, if they arc
projected obliquely ;
so from the supposed existence of a force directed to
any centre, it will follow, by the like necessity, that all bodies upon which
this force acts mast either descend directly to that centre, or at least devi
ate perpetually towards it from right lines, if otherwise they should have
moved obliquely in these right lines.
And how from the motions given we may infer the forces, or from the
forces given we may determine the motions, is shewn in the two first Books
of our Principles of Philosophy.
If the earth is supposed to stand still, and the fixed stars to be revolved
in free spaces in the space of 24 hours, it is certain the forces by which
the fixed stars are retained in their orbs are not directed to the earth, but
to the centres of the several orbs, that is, of the several parallel circles,
which the fixed stars, declining to one side and the other from the equator,
describe daily ;
also that by radii drawn to the centres of those orbs tht
fixed stars describe areas exactly proportional to the times of description.
Then, because the periodic times are equal (by Cor. Ill, Prop. IV, Book 1),
it follows that the centripetal forces are as the radii of the several orbs,
and that they will perpetually revolve in the same orbs. And the like
consequences may be drawn from the supposed diurnal motion of the
planets.
That forces should be directed to no body on which they physically de
pend, but to innumerable imaginary points in the axis of the earth, is an
hypothesis too incongruous. It is more incongruous still that those forces
should increase exactly in proportion of the distances from this axis ; for
THE SYSTEM OF THE WORLD. 515
this is an indi ation of an increase to immensity, or rather to infinity ;
whereas the forces of natural things commonly decrease in receding from
the fountain from which they flow. But, what is yet more absurd, neither
are the areas described by the same star proportional to the times, nor are
its revolutions performed in the same orb ;
for as the star recedes from the
neighbouring pole, both areas and orb increase; and from the increase of
the urea it is demonstrated that the forces are not directed to the axis of
the earth. And this difficulty (Cor. 1, Prop. II) arises from the twofold
motion that is observed in the fixed stars, one diurnal round the axis of
the earth, the other exceedingly slow round the axis of the ecliptic. And
the explication thereof requires a composition of forces so perplexed and
so variable, that it is hardly to be reconciled with any physical theory.
That there are centripetal forces actually directed to the bodies of the
sun, of the earth, and other planets, I thus infer.
The moon revolves about our earth, and by radii drawn to its centre
(p. 390) describes areas nearly proportional to the times in which they are
described, as is evident from its velocity compared with its apparent diame
ter
;
for its motion is slower when its diameter is less (and therefore its
distance greater), and its motion is swifter when its diameter is greater.
The revolutions of the satellites of Jupiter about that planet are more
regular (p. 386) : for they describe circles concentric with Jupiter by equa
ble motions, as exactly as our senses can distinguish.
And so the satellites of Saturn are revolved about this planet with mo
tions nearly (p. 387) circular and equable, scarcely disturbed by any eccen
tricity hitherto observed.
That Venus and Mercury are revolved about the sun, is demonstrable
from their moon-like appearances (p. 388) . when they shine with a full
face, they are in those parts of their orbs which in respect of the earth lie
beyond the sun ; when they appear half full, they are in those parts whicli
Ire over against the sun ; when horned, in those parts which lie between
the earth and the sun ; and sometimes they pass over the sun s disk, when
directly interposed between the eirth and the sun.
And Venus, with a motion almost uniform, describes an orb nearly cir
cular and concentric with the sun.
But Mercury, with a more eccentric motion, makes remarkable ap
proaches to the sun, and goes off again by turns ; but it is always swifter
as it is near to the sun, and therefore by a radius drawn to the sun still
describes areas proportional to the times.
Lastly, that the earth describes about the sun, or the sun about the
earth, by a radius from the one to the other, areas exactly proportional to
the times, is demonstrable from the apparent diameter of the sun com
pared with its apparent motion.
These are astronomical experiments ; from which it follows, by Prop. I,
516 THE SYSTEM OF THE WORLD.
11, III, in the first Book of our Pn/triples, and their Corollaries (p.
213, 214). that there are centripetal forces actually directed (either accu
rately or without considerable error) to the centres of the earth, of Jupi
ter, of S.iturn, and of the sun. In Mercury, Venus, Mars, and the lesser
planets, wheie experiments are wanting, the arguments from analogy must
be allowed in their place.
That those forces (p. 212, 213, 214) decrease in the duplicate propor
tion of the distances from the centre of every planet, appears by Cor. VI,
Prop. IV, Book 1
;
for the periodic times of the satellites of Jupiter are
one to another (p. 386, 387) in the sesquiplicate proportion of their dis
tances from the centre of this planet.
This proportion has been long ago observed in those satellites ; and Mr.
Flamsted, who had often measured their distances from Jupiter by the
micrometer, and by the eclipses of the satellites, wrote to me, that it holds
to all the accuracy that possibly can be discerned by our senses. And he
sent me the dimensions of their orbits taken by the micrometer, a*nd re
duced to the mean distance of Jupiter from the earth, or from the sun,
together with the times of their revolutions, as follows :
Wherce the sesquiplicate proportion may be easily seen. For example ;
the 16(f 18h
. 05 13" is to the time l d
. 18h
. 28 36" as 493i" x V 493i"
to 108 X V 108", neglecting those small fractions which, in observing,
cannot ./e certainly determined.
Befo e the invention of the micrometer, the same distances vrere deter
mined 7 \ semi-diameters of Jupiter thus :
After the invention of the micrometer :
THE SYSTEM OF THE WORLD. 6l7
And the periodic times of those satellites, by the observations of Mr.
Flamsted, are l d
. 18h
. 28 36"
|
3(l
. 13". 17 54"
| 7(1
. 3h
. 59 36"
|
16".
IS11
. 5 13". as above.
And the distances thence computed are 5,578 | 8,878 | 14,168 | 24,968,
accurately agreeing with the distances by observation.
Cassini assures us (p. 388, 389) that the same proportion is observed
in the circum-saturnal planets. But a longer course of observations is
required before we can have a certain and accurate theory of those planets.
In the circum-solar planets, Mercury and Venus, the same proportion
holds with great accuracy, according to the dimensions of their orbs, as
determined by the observations of the best astronomers.
That Mars is revolved about the sun is demonstrated from the phases
which it shews, and the proportion of its apparent diameters (p. 388, 389,
and 390) ; for from its appearing fall near conjunction with the sun, and
gibbous in its quadratures, it is certain that it surrounds the sun.
And since its diameter appears about five times greater when in opposi
tion to the sun than when in conjunction therewith, and its distance from
the earth is reciprocally as its apparent diameter, that distance will be
about five times less when in opposition to than when in conjunction with
the sun; but in both cases its distance from the sun will be nearly about
the same with the distance which is inferred from its gibbous appearance
in the quadratures. And as it encompasses the sun at almost equal dist nces,
but in respect of the earth is very unequally distant, so by radii drawn
to the sun it describes areas nearly uniform ; but by radii drawn to the
earth, it is sometimes swift, sometimes stationary, and sometimes retrograde.
That Jupiter, in a higher orb than Mars, is likewise revolved about the
sun, with a motion nearly equable, as well in distance as in the areas des
cribed, 1 infer thus.
Mr. Flamsted assured me, by letters, that all the eclipses of the inner
most satellite which hitherto have been well observed do agree with his
theory so nearly, as never to differ therefrom by two minutes of time ;
that in the outmost the error is little greater ;
in the outmost but one,
scarcely three times greater ;
that in the innermost but one the difference
is indeed much greater, yet so as to agree as nearly with his computation?
as the moon does with the common tables ; and that he computes those
eclipses only from the mean motions corrected by the equation of light dis
covered and introduced by Mr. Rower. Supposing, then, that the theory
differs by a less error than that of 2 from the motion of the outmost sat
ellite as hitherto described, and taking as the periodic time 16 1
. 18h
. 5 13"
to 2 in time, so is the whole circle or 360 to the arc 1
48", the error ol
Mr. Flamsted s computation, reduced to the satellite s orbit, will be less
than 1 48"
;
that is, the longitude of the satellite, as seen from tlie centre
of Jupiter; will be determined with a less error than 1 48". But when
518 THE SYSTEM OF THE WORLD.
the satellite is in the middle of the shadow, that longitude is the same with
the heliocentric longitude of Jupiter ; and, therefore, the hypothesis which
Mr. Flamsted follows, viz., the Copernican, as improved by Kepler, and
fas to the motion of Jupiter) lately corrected by himself, rightly represents
that longitude within a less error than 1 48"
; but by this longitude, to
gether with the geocentric longitude, which is always easily found, the dis
tance of Jupiter from the sun is determined ; which must, therefore, be the
very same with that which the hypothesis exhibits. For that greatest error
of I 48" that can happen in the heliocentric longitude is almost insensi
ble, and quite to be neglected, and perhaps may arise from some yet undis
covered eccentricity of the satellite : but since both longitude and distance
are rightly determined, it follows of necessity that Jupiter, by radii drawn
to the sun. describes areas so conditioned as the hypothesis requires, that is.
proportional to the times.
And the same thing may be concluded of Saturn from his satellite, by
the observations of Mr. Huygens and Dr. Halley ; though a longer series
of observations is yet wanting to confirm the thing, and to bring it under
a sufficiently exact computation.
For if Jupiter was viewed from the sun, it would never appear retro
grade nor stationary, as it is seen sometimes from the earth, but always to
go forward with a motion nearly uniform (p. 389). And from the very
great inequality of its apparent geocentric motion, we infer (by Prop. Ill
Cor. IV) that the force by which Jupiter is turned out of a rectilinear course,
and made to revolve in an orb, is not directed to the centre of the earth.
And the same argument holds good in Mars and in Saturn. Another centre
of these forces is therefore to be looked for (by Prop. II and III, and the
Corollaries of the latter), about which the areas described by radii inter
vening may be equable ; and that this is the sun, we have proved already
in Mars and Saturn nearly, but accurately enough in Jupiter. It may be
alledged that the sun and planets are impelled by some other force equally
and in the direction of parallel lines ; but by such a force (by Cor. VI of
the Laws of Motion) no change would happen in the situation of the
planets one to another, nor any sensible eifect follow : but our business is
with the causes of sensible effects. Let us, therefore, neglect every such
force as imaginary and precarious, and of no use in the phenomena of the
heavens ; and the whole remaining force by which Jupiter is impelled will
be directed (by Prop. Ill, Cor. I) to the centre of the sun.
The distances of the planets from the sun come out the same, whether,
with Tycho, we place the earth in the centre of the system, or the sun with
Copernicus : and we have already proved that these distances are true ir.
Jupiter.
Kepler and Bullialdiis have, with great care (p. 388), determined the
listances of the planets from the sun ; and hence it is that their table.-?
THE SYSTEM OF THE WORLD. 519
agree best with the heavens. And in all the planets, in Jupiter and Mars,
in Saturn and the earth, as well as in Venus and Mercury, the cubes of their
distances are as the squares of their periodic times ; and therefore (by Cor.
VI, Prop. IV) the centripetal circum-solar force throughout all the plane
tary regions decreases in the duplicate proportion of the distances from the
sun. In examining this proportion, we are to use the mean distances, or
the transverse semi-axes of the orbits (by Prop. XV). arid to neglect those
little fractions, which, in denning the orbits, may have arisen from the in
sensible errors of observation, or may be ascribed to other causes which we
shall afterwards explain. And thus we shall always find the said propor
tion to hold exactly; for the distances of Saturn, Jupiter, Mars, the Earth,
Venus, and Mercury, from the sun, drawn from the observations of as
tronomers, are, according to the computation of Kepler, as the numbers
95 LOGO, 519650, 152350, 100000, 72400, 3S806; by the computation of
/iHllialdus, as the numbers 95419S, 522520, 152350, 100000, 72393,
38585 ; and from the periodic times they come out 953806, 520116, 152399,
100000, 72333, 38710. Their distances, according to Kepler and
Ktillwldus, scarcely differ by any sensible quantity, and where they
differ most the distances drawn from the periodic times, fall in between them.
That the circum-terrestrial force likewise decreases in the duplicate pro
portion of the distances, I infer thus.
The mean distance of the moon from the centre of the earth, is, in semidiameters
of the earth, according to Ptolemy, Kepler in his Ephemerides,
Bidliuldus, Hevelius, and Ricciolns, 59 ; according to Flamsted, 59| ;
according to Tycho, 56 1
;
to Vendelin, 60 ; to Copernicus, 60 1 : to Kircher,
62i (p. 391, 392, 393).
Cut Tycho, and all that follow his tables of refraction, making the
refractions of the sun and moon (altogether against the nature of light)
to exceed those of the fixed stars, and that by about four or five minutes
in the horizon, did thereby augment the horizontal parallax of the moon
by about the like number of minutes ; that is, by about the 12th or 15th
part of the whole parallax. Correct this error, and the distance will be
come 60 or 61 semi-diameters of the earth, nearly agreeing with what
others have determined.
Let us, then, assume the mean distance of the moon 60 semi-diameters
of the earth, and its periodic time in respect of the fixed stars 27d
. 7h
. 43 ,
as astronomers have determined it. And (by Cor. VI, Prop. IV) a body
revolved in our air, near the surface of the earth supposed at rest, by
means of a centripetal force which should be to the same force at the dis
tance of the moon in the reciprocal duplicate proportion of the distances
from the centre of the earth, that is, as 3600 to 1, would (secluding the
resistance of the air) complete a revolution in l
h
. 24 27".
Suppose the circumference of the earth to be 123249600 Paris feet; ar
52C THE SYSTEM OF THE WORLD.
has been determined by the late mensuration of the French (vide p. 406) ;
then the sume body, deprived of its circular motion, and falling by the
impulse of the same centripetal force as before, would, in one second of
time, describe 15-^ Paris feet.
This we infer by a calculus formed upon Prop. XXXYI, and it agrees
with what we observe in all bodies about the earth. For by the experi
ments of pendulums, and a computation raised thereon, Mr. Hnygens has
demonstrated that bodies falling by all that centripetal force with which
(of whatever nature it is) they are impelled near the surface of the earth,
do, in one second of time, describe 15 T^ Paris feet.
But if the earth is supposed to move, the earth and moon together (by
Cor. IV of the Laws of Motion, and Prop. LVID will be revolved about
their common centre of gravity. Ana the moon (by Prop. LX) will in
the same periodic time, 27 1
. 7h
. 43 , with the same circum terrestrial force
diminished in the duplicate proportion of the distance, describe an orbit
whose semi-diameter is to the semi-diameter of the former orbit, that is, to
60 semi-diameters of the earth, as the sum of both the bodies of the earth
and moon to the first of two mean proportionals between this sum and the
body of the earth
;
that is, if we suppose the moon (on account of its
mean apparent diameter 31^ ) to be about ^ of the earth, as 43 to
^ 42 a-
43|2, or as about 128 to 127. And therefore the semi-diameter
of the orbit, that is, the distance between the centres of the moon and
earth, will in this case be 60^ semi-diameters of the earth, almost the same
with that assigned by Copernicus, which the Tychonic observations by no
means disprove ; and, therefore, the duplicate proportion of the decrement
of the force holds good in this distance. I have neglected the increment
of the orbit which arises from the action of the sun as inconsiderable ;
but if that is subducted, the true distance will remain about 60|- semidiameters
of the earth.
But farther (p. 390) ; this proportion of the decrement of the forces is
confirmed from the eccentricity of the planets, and the very slow motion
of their apses ;
for (by the Corollaries of Prop. XLV) in no other proportion
could the circum-solar planets once in every revolution descend to
their least and once ascend to their greatest distance from the sun, and the
places of those distances remain immoveable. A small error from the du
plicate proportion would produce a motion of the apses considerable in
every revolution, but in many enormous.
But now, after innumerable revolutions, hardly any such motion ha&
been perceived in the orbs of the circum-solar planets. Some astronomers
affirm that there is no such motion; others reckon it no greater than what
may easily arise from the causes hereafter to be assigned, and is of no mo
ment in the present question.
THE SYSTEM 01 THE WORLD. 521
We may even neglect the motion of the moon s apsis (p. 390, 391), which
is far greater than in the circum-solar planets, amounting in every revolu
tion to three degrees ; and from this motion it is demonstrable that the
circum-terrestrial force decreases in no less than the duplicate, but far less
than the triplicate proportion of the distance ; for if the duplicate propor
tion was gradually changed into the triplicate, the motion of the apsis
would thereby increase to infinity; and, therefore, by a very small muta
tion, would exceed the motion of the moon s apsis. This slow motion arises
from the action of the circum-solar force, as we shall afterwards explain.
But, secluding this cause, the apsis or apogeon of the moon will be fixed,
and the duplicate proportion of the decrease of the circum-terrestrial force
in different distances from the earth will accurately take place.
Now that this proportion has been established, we may compare the
forces of the several planets among themselves (p. 391).
In the mean distance of Jupiter from the earth, the greatest elongation
of the outmost satellite from Jupiter s centre (by the observations of Mr.
Flamsted] is 8 13"
; and therefore the distance of the satellite from the
centre of Jupiter is to the mean distance of Jupiter from tne centre of the
sun as 124 to 52012, but to the mean distance of Venus from the centre
of the sun as 124 to 7234; and their periodic times are 16 d
. and 224f
d
;
and from hence (according to Cor. II, Prop. IV), dividing the distances by
the squares of the times, we infer that the force by which the satellite is
impelled towards Jupiter is to the force by which Venus is impelled to
wards the sun as 442 to 143 ; and if we diminish the force by which the
satellite is impelled in the duplicate proportion of the distance 124 to
7234, we shall have the circum-jovial force in the distance of Venus from
the sun to the circum-solar force by which Venus is impelled as yW to
143, or as 1 to 1100; wherefore at equal distances the circum-solar force
is 1100 times greater than the circum-jovial.
And, by the like computation, from the periodic time of the satellite ot
Saturn 15(l
. 22h
. and its greatest elongation from Saturn, while that planet
is in its mean distance from us, 3 20",
it follows that the distance of this
satellite from Saturn s centre is to the distance of Venus from the sun as
92| to 7234; and from thence that the absolute circum-solar force is 2360
times greater than the absolute circum-saturnal.
From the regularity of the heliocentric and irregularity of the geocen
tric motions of Venus, of Jupiter, and the other planets, it is evident (by
Cor. IV, Prop. Ill) that the circum-terrestrial force, compared with the cir
cum-solar, is very small.
Ricciolus and Vendelin have severally tried to determine the sun s par
allax from the moon s dichotomies observed by the telescope, and they agree
that it does not exceed half a minute.
Kepler, from Ti/cho s observations and his own, found the parallax of
o22 THE SYSTEM OF THE WORLD.
Mars insensible, even in opposition to the sun, when that parallax is some
thing greater than the sun s.
Flamsted attempted the same parallax with the micrometer in the perigeon
position of Mars, but never found it above 25"
; and thence conclud
