to be moved.
PROPOSITION XIII. THEOREM XIII.
The planets move in ellipses tvhicli have their common focus in the centre
of the sini ; and, by radii drawn, to tJtat centre, they describe areas pro
portional to the times of description.
We have discoursed above of these motions from the Phenomena. Now
that we know the principles on which they depend, from those principles
we deduce the motions of the heavens a priori. Because the weights of
the planets towards the sun are reciprocally as the squares of their distan
ces from the sun s centre, if the sun was at rest, and the other planets did
not mutually act one upon another, their orbits would be ellipses, having
the sun in their common focus; and they would describe areas proportional
to the times of description, by Prop. I and XI, and Cor. 1, Prop. XIII,
Book I. But the mutual actions of the planets one upon another are so
very small, that they may be neglected ; and by Prop. LXVI, Book I, they
less disturb the motions of the planets around the sun in motion than if
those motions were performed about the sun at rest.
It is true, that the action of Jupiter upon Saturn is not to be neglected;
for the force of gravity towards Jupiter is to the force of gravity towards
the sun (at equal distances, Cor. 2, Prop. VIII) as 1 to 1067; and therefore
in the conjunction of Jupiter and Saturn, because the distance of Saturn
from Jupiter is to the distance of Saturn from the sun almost as 4 to 9, the
gravity of Saturn towards Jupiter will be to the gravity of Saturn towards
the sun as 81 to 16 X 1067; or, as 1 to about 21 1. And hence arises a
perturbation of the orb of Saturn in every conjunction of this planet with
Tupiter, so sensible, that astronomers are puzzled with it. As the planet
BOOK III.] OF NATURAL PHILOSOPHY. 403
is differently situated in these conjunctions, its eccentricity is sometimes
augmented, sometimes diminished; its aphelion is sometimes carried for
ward, sometimes backward, and its mean motion is by turns accelerated and
retarded ; yet the whole error in its motion about the sun, though arising
from so great a force, may be almost avoided (except in the mean motion)
by placing the lower focus of its orbit in the common centre of gravity of
Jupiter and the sun (according to Prop. LXVII, Book I), and therefore that
error, when it is greatest, scarcely exceeds two minutes ; and the greatest
error in the mean motion scarcely exceeds two minutes yearly. But in the
conjunction of Jupiter and Saturn, the accelerative forces of gravity of the
sun towards Saturn, of Jupiter towards Saturn, and of Jupiter towards the
sun, are almost as 16, 81, and -
~o^~~ >
or 156609: and therefore
the difference of the forces of gravity of the sun towards Saturn, and of
Jupiter towards Saturn, is to the force of gravity of Jupiter towards the
sun as 65 to 156609, or as 1 to 2409. But the greatest power of Saturn
to disturb the motion of Jupiter is proportional to this difference; and
therefore the perturbation of the orbit of Jupiter is much less than that of
Saturn s. The perturbations of the other orbits are yet far less, except that
the orbit of the earth is sensibly disturbed by the moon. The common
centre of gravity of the earth and moon moves in an ellipsis about the sun
in the focus thereof, and, by a radius drawn to the sun, describes areas pro
portional to the times of description. But the earth in the mean time by
a menstrual motion is revolved about this common centre.
PROPOSITION XIV. THEOREM XIV.
The aphelions and nodes of the orbits of the planets are fixed.
The aphelions are immovable by Prop. XI, Book I
; and so are the
planes of the orbits, by Prop. I of the same Book. And if the planes are
fixed, the nodes must be so too. It is true, that some inequalities may
arise from the mutual actions of the planets and comets in their revolu
tions
; but these will be so small, that they may be here passed by.
COR. 1. The fixed stars are immovable, seeing they keep the same posi
tion to the aphelions and nodes of the planets.
COR. 2. And since these stars are liable to no sensible parallax from the
annual motion of the earth, they can have no force, because of their im
mense distance, to produce any sensible effect in our system. Not to
mention that the fixed stars, every where promiscuously dispersed in the
heavens, by their contrary attractions destroy their mutual actions, by
Prop. LXX, Book I.
SCHOLIUM.
Since the planets near the sun (viz. Mercury, Venus, the Earth, and
404 THE MATHEMATICAL PRINCIPLES [B .-OK IIL
Mars) are so small that they can act with but little force upon each other,
therefore their aphelions and nodes must be fixed, excepting in so far as
they are disturbed by the actions of Jupiter and Saturn, and other higher
bodies. And hence we may find, by the theory of gravity, that their aphe
lions move a little in consequentw, in respect of the fixed stars, and that
in the sesqui plicate proportion of their several distances from the sun. So
that if the aphelion of Mars, in the space of a hundred years, is carried
33 20" in consequent-la, in respect of the fixed stars, the aphelions of the
Earth, of Venus, and of Mercury, will in a hundred years be carried for
wards 17 40", 10 53 , and 4 16", respectively. But these motions are
so inconsiderable, that we have neglected them in this Proposition,
PROPOSITION XV. PROBLEM I.
To find the principal diameters
<>f
the orbits of the planets.
They are to be taken in the sub-sesquiplicate proportion of the periodic
times, by Prop. XV, Book I, and then to be severally augmented in the
proportion of the sum of the masses of matter in the sun and each planet
to the first of two mean proportionals betwixt that sum and the quantity of
matter in the sun, by Prop. LX, Book I.
PROPOSITION XVI. PROBLEM II.
To find the eccentricities and aphelions of the planets.
This Problem is resolved by Prop. XVIII, Book I.
PROPOSITION XVII. THEOREM XV.
That the diurnal motions of the planets are uniform, and that the
libration of the moon arises from its diurnal motion.
The Proposition is proved from the first Law of Motion, and Cor. 22,
Prop. LXVI, Book I. Jupiter, with respect to the fixed stars, revolves in
9 1
. 5(5
; Mars in 24h
. 39 ; Venus in about 23h
.
; the Earth in 23 1
. 56 ; the
Sun in 25 1
days, and the moon in 27 days, 7 hours, 43 . These things
appear by the Phasnomena. The spots in the sun s body return to the
same situation on the sun s disk, with respect to the earth, in 27 days ; and
therefore with respect to the fixed stars the sun revolves in about 25|days.
But because the lunar day, arising from its uniform revolution about its
axis, is menstrual, that is, equal to the time of its periodic revolution in
its orb, therefore the same face of the moon wr ill be always nearly turned to
the upper focus of its orb ; but, as the situation of that focus requires, will
deviate a little to one side and to the other from the earth in the lower
focus j and this is the libration in longitude ; for the libration in latitude
arises from the moon s latitude, and the inclination of its axis to the plane
of the ecliptic. This theory of the libration of the moon, Mr. N. Mercato*
BOOK III.] OF NATURAL PHILOSOPHY. 4()
in his Astronomy, published at the beginning of the year 1676. explained
more fully out of the letters I sent him. The utmost satellite of Saturn
eeems to revolve about its axis with a motion like this of the moon, respect
ing Saturn continually with the same face; for in its revolution round
Saturn, as often as it comes to the eastern part of its orbit, it is scarcel)
visible, and generally quite disappears ; which is like to be occasioned by
some spots in that part of its body, which is then turned towards the earth,
as M. Cassini has observed. So also the utmost satellite of Jupiter seema
to revolve about its axis with a like motion, because in that part of its body
which is turned from Jupiter it has a spot, which always appears as if it
were in Jupiter s own body, whenever the satellite passes between Jupiter
and our eye.
PROPOSITION XVIII. THEOREM XVI.
That the axes of the planets are less than the diameters drawn perpen
dicular to the axes.
The equal gravitation of the parts on all sides would give a spherical
figure to the planets, if it was not for their diurnal revolution in a circle.
By that circular motion it comes to pass that the parts receding from the
axis endeavour to ascend about the equator ; and therefore if the matter is
in a fluid state, by its ascent towards the equator it will enlarge the di
ameters there, and by its descent to wards the poles it will shorten the axis.
So the diameter of Jupiter (by the concurring observations of astronomers)
is found shorter betwixt pole and pole than from east to west. And, by
the same argument, if our earth was not higher about the equator than at
the poles, the seas would subside about the poles, and, rising toward* Ikf
equator, would lay all things there under water.
PROPOSITION XIX. PROBLEM III
Tofind the proportion of the axis of a planet to the dia meter j j*,rpendici/
lar thereto.
Our countryman, Mr. Norwood, measuring a distance of 005751 feet of
London measure between London and YorA:, in 1635, and obs,-rvino- the
difference of latitudes to be 2 28 , determined the measure of one degree
to be 3671 96 feet of London measure, that is 57300 Paris toises. M
Picart, measuring an arc of one degree, and 22 55" of the meridian be
tween Amiens and Malvoisine, found an arc of one degree to be 57060
Paris toises. M. Cassini, the father, measured the distance upon the me
ridian from the town of Collionre in Roussillon to the Observatory of
Pari; and his son added the distance from the Observatory to the Cita
del of Dunkirk. The whole distance was 486156^ toises and the differ
ence of the latitudes of Collionre and Dunkirk was 8 degrees, and 31
106 THE MATHEMATICAL PRINCIPLES [BOOK 1IJ.
llf". Hence an arc of one degree appears to be 57061 Paris toises.
And from these measures we conclude that the circumference of the earth
is 123249600, and its semi-diameter 19615800 Paris feet, upon the sup
position that the earth is of a spherical figure.
In the latitude of Paris a heavy body falling in a second of time de
scribes 15 Paris feet, 1 inch, 1 J line, as above, that is, 2173 lines J. The
weight of the body is diminished by the weight of the ambient air. Let
us suppose the weight lost thereby to be TT ^o-o- Par^ ^ ^he whole weight ;
then that heavy body falling in, vacua will describe a height of 2174 lines
in one second of time.
A body in every sidereal day of 23 1
. 56 4" uniformly revolving in a
circle at the distance of 19615SOO feet from the centre, in one second oi
time describes an arc of 1433,46 feet
;
the versed sine of which is 0,0523656 1
feet, or 7,54064 lines. And therefore the force with which bodies descend
in the latitude of Paris is to the centrifugal force of bodies in the equator
arising from the diurnal motion of the earth as 2174 to 7,54064.
The centrifugal force of bodies in the equator is to the centrifugal force
with which bodies recede directly from the earth in the latitude of Parin
48 50 10" in the duplicate proportion of the radius to the cosine of the
latitude, that is, as 7,54064 to 3,267. Add this force to the force with
which bodies descend by their weight in the latitude of Paris, and a body,
in the latitude of Paris, falling by its whole undiminished force of gravity,
in the time of one second, will describe 2177,267 lines, or 15 Paris feet,
1 inch, and 5,267 lines. And the total force of gravity in that latitude
will be to the centrifugal force of bodies in the equator of the earth as
2177,267 to 7,54064, or as 289 to 1.
Wherefore if APBQ, represent the figure of the
earth, now no longer spherical, but generated by the
rotation of an ellipsis about its lesser axis PQ, ; and
ACQqca a canal full of water, reaching from the pole
Qq to the centre Cc, and thence rising to the equator
Art ; the weight of the water in the leg of the canal
ACca will be to the weight of water in the other leg
QCcq as 289 to 288, because the centrifugal force arising from the circu
lar motion sustains and takes off one of the 289 parts of the weight (in the
one leg), and the weight of 288 in the other sustains the rest. But by
computation (from Cor. 2, Prop. XCI, Book I) I find, that, if the matter
of the earth was all uniform, and without any motion, and its axis PQ,
were to the diameter AB as 100 to 101, the force of gravity in the
place Q towards the earth would be to the force of gravity in the same
place Q towards a sphere described about the centre C with the radius
PC, or QC, as 126 to 125. And, by the same argument, the force of
gravity in the place A towards the spheroid generated by the rotation of
BOOK III.] OF NATURAL PHILOSOPHY. 407
the ellipsis APBQ, about the axis AI3 is to the force of gravity in the
same place A, towards the sphere described about the centre C with the
radius AC, as 125 to 126. But the force of gravity in the place A to
wards the earth is a mean proportional betwixt the forces of gravity to
wards the spheroid and this sphere; because the sphere, by having its di
ameter PQ, diminished in the proportion of 101 to 100, is transformed into
the figure of the earth ; and this figure, by having a third diameter per
pendicular to the two diameters AB and PQ, diminished in the same pro
portion, is converted into the said spheroid ; and the force of gravity in A,
in either case, is diminished nearly in the same proportion. Therefore the
force of gravity in A towards the sphere described about the centre C with
the radius AC, is to the force of gravity in A towards the earth as 126 to
1251. And the force of gravity in the place Q towards the sphere de
scribed about the centre C with the radius QC, is to the force of gravity
in the place A towards the sphere described about the centre C, with the
radius AC, in the proportion of the diameters (by Prop. LXXII, Book I),
that is, as 100 to 101. If, therefore, we compound those three proportions
126 to 125, 126 to 125|. and 100 to 101, into one, the force of gravity in
the place Q towards the earth will be to the force of gravity in the place
A towards the earth as 126 X 126 X 100 to 125 X 125| X 101 ; or as
:>01 to 500.
Now since (by Cor. 3, Prop. XCI, Book I) the force of gravity in either
leg of the canal ACca, or QCcy, is as the distance of the places from the
centre of the earth, if those legs are conceived to be divided by transverse.,
parallel, and equidistant surfaces, into parts proportional to the wholes,
the weights of any number of parts in the one leg ACca will be to the
weights of the same number of parts in the other leg as their magnitudes
and the accelerative forces of their gravity conjunctly, that is, as 10 J to
100, and 500 to 501. or as 505 to 501. And therefore if the centrifugal
force of every part in the leg ACca, arising from the diurnal motion, was
to the weight of the same part as 4 to 505, so that from the weight of
every part, conceived to be divided into 505 parts, the centrifugal force
might take off four of those parts, the weights would remain equal in each
leg, and therefore the fluid would rest in an equilibrium. But the centri
fugal force of every part is to the weight of the same part as 1 to 289 ;
that is, the centrifugal force, which should be T y parts of the weight, is
only |g part thereof. And, therefore, I say, by the rule of proportion,
that if the centrifugal force j ^ make the height of the water in the leg
ACca to exceed the height of the water in the leg QCcq by one T | part
of its whole height, the centrifugal force -^jj will make the excess of the
height in the leg ACca only ^{^ part of the height of the water in the
other leg QCcq ; and therefore the diameter of the earth at the equator, is
to its diameter from pole to pole as 230 to 229. And since the mean semi
108 THE MATHEMATICAL PRINCIPLES [BooK III.
diameter of the earth, according to PicarVs mensuration, is 19615800
Paris feet, or 3923,16 miles (reckoning 5000 feet to a mile), the earth
will be higher at the equator than at the poles by 85472 feet, or 17^-
miles. And its height at the equator will be about 19658600 feet, and at
the poles 19573000 feet.
If, the density and periodic time of the diurnal revolution remaining the
same, the planet was greater or less than the earth, the proportion of the
centrifugal force to that of gravity, and therefore also of the diameter be
twixt the poles to the diameter at the equator, would likewise remain the
game. But if the diurnal motion was accelerated or retarded in any pro
portion, the centrifugal force would be augmented or diminished nearly in
the same duplicate proportion ; and therefore the difference of the diame
ters will be increased or diminished in the same duplicate ratio very nearly.
And if the density of the planet was augmented or diminished in any pro
portion, the force of gravity tending towards it would also be augmented
or diminished in the same proportion : and the difference of the diameters
contrariwise would be diminished in proportion as the force of gravity is
augmented, and augmented in proportion as the force of gravity is dimin
ished. Wherefore, since the earth, in respect of the fixed stars, revolves in
23h
. 56 , but Jupiter in 9h
. 56 , and the squares of their periodic times are
as 29 to 5, and their densities as 400 to 94 , the difference of the diameters
29 400 1
of Jupiter will be to its lesser diameter as X ^^ X ^Tm
to 1; or as 1 to
9 f, nearly. Therefore the diameter of Jupiter from east to west is to its
diameter from pole to pole nearly as 10 to 9|-. Therefore since its
greatest diameter is 37", its lesser diameter lying between the poles will
be 33" 25" . Add thereto about 3 for the irregular refraction of light,
and the apparent diameters of this planet will become 40 and 36" 25"
;
which are to each other as 11 -j
to 10^, very nearly. These things are so
upon the supposition that the body of Jupiter is uniformly dense. But
now if its body be denser towards the plane of the equator than towards
the poles, its diameters may be to each other as 12 to 11, or 13 to 12, or
perhaps as 14 to 13.
And Cassini observed in the year 1691, that the diameter of Jupiter
reaching from east to west is greater by about a fifteenth part than the
other diameter. Mr. Pound with his 123 feet telescope, and an excellent
micrometer, measured the diameters of Jupiter in the year 1719, and found
them as follow.
K HI. OF NATURAL PHILOSOPHY. 409
So thut the theory agrees with the phenomena ;
for the planets are more
heated by the sun s rays towards their equators, and therefore are a lit fie
more condensed by that heat than towards their poles.
Moreover, that there is a diminution of gravity occasioned by the diur
nal rotation of the earth, and therefore the earth rises higher there than it
does at the poles (supposing that its matter is uniformly dense), will ap
pear by the experiments of pendulums related under the following Propo
sition.
PROPOSITION XX. PROBLEM IV.
Tofind and compare together the weights of bodies in the different re
gions of our earth.
Because the weights of the unequal legs of the canal
of water ACQqca are equal ; and the weights of the
parts proportional to the whole legs, and alike situated
in them, are one to another as the weights of the P|
wholes, and therefore equal betwixt themselves ; the
weights of equal parts, and alike situated in the legs,
will be reciprocally as the legs, that is, reciprocally as
230 to 229. And the case is the same in all homogeneous equal bodies alike
situated in the legs of the canal. Their weights are reciprocally as the legs,
that is, reciprocally as the distances of the bodies from the centre of the earth.
Therefore if the bodies are situated in the uppermost parts of the canals, or on
the surface of the earth, their weights will be one to another reciprocally as
their distances from the centre. And. by the same argument, the weights in
all other places round the whole surface of the earth are reciprocally as the
distances of the places from the centre ; and, therefore, in the hypothesis
of the earth s being a spheroid are given in proportion.
Whence arises this Theorem, that the increase of weight in passing from
tne equator to the poles is nearly as the versed sine of double the latitude ;
or, which comes to the same thinir, as the square of the right sine of the
latitude ; and the arcs of the degrees of latitude in the meridian increase
nearly in the same proportion. And, therefore, since the latitude of Paris
is 48 50 , that of places under the equator 00 00 , and that of places
under the poles 90 ; and the versed sines of double those arcs are
11334,00000 and 20000, the radius being 10000 ; and the force of gravity
at the pole is to the force of gravity at the equator as 230 to 229 ; and
the excess of the force of gravity at the pole to the force of gravity at the
equator as 1 to 229 ; the excess of the force of gravity in the latitude of
Paris will be to the force of gravity at the equator as 1 X Htll to 229,
or as 5667 to 2290000. And therefore the whole forces of gravity in
those places will be one to the other as 2295667 to 2290000. Wherefore
since the lengths of pendulums vibrating in equal times are as the forces of
410 THE MATHEMATICAL PRINCIPLES [BOOK III.
gravity, and in the latitude of Paris, the length of a pendulum vibrating
seconds is 3 Paris feet, and S lines, or rather because of the weight of
the air, 8f lines, the length of a pendulum vibrating in the same time
arider the equator will be shorter by 1,087 lines. And by a like calculus
the following table is made.
By this table, therefore, it appears that the inequality of degrees is sc
small, that the figure of the earth, in geographical matters, may be con
sidered as spherical ; especially if the earth be a little denser towards the
plane of the equator than towards the poles.
Now several astronomers, sent into remote countries to make astronomical
observations, have found that pendulum clocks do accordingly move slower
near the equator than in our climates. And, first of all, in the year I 72,
M. Richer took notice of it in the island of Cayenne ; for when, in the
month of August, he was observing the transits of the fixed stars over the
meridian, he found his clock to go slower than it ought in respect of the
mean motion of the sun at the rate of 2 29" a day. Therefore, fitting up
a simple pendulum to vibrate in seconds, which were measured by an ex
cellent clock, he observed the length of that simple pendulum ; and this he
did over and over every week for ten months together. And upon his re
turn to France, comparing the length of that pendulum with the length
iiJ.j OF NATURAL PHILOSOPHY. 411
of the pendulum at Paris (which was 3 Paris feet and 8f lines), he found
it shorter by 1 j line.
Afterwards, our friend Dr. Halley, about the year 1677, arriving at the
island of St. Helena, found his pendulum clock to go slower there than at
Isondon without marking the difference. But he shortened the rod of
his clock by more than the \ of an inch, or l line
; and to effect this, be
cause the length of the screw at the lower end of the rod was riot sufficient,
he interposed a wooden ring betwixt the nut and the ball.
Then, in the year 1682, M. Varin and M. des Hayes found the length
of a simple pendulum vibrating in seconds at the Royal Observatory of
Paris to be 3 feet and S| lines. And by the same method in the island
of Goree, they found the length of an isochronal pendulum to be 3 feet and
6 1 lines, differing from the former by two lines. And in the same year,
going to the islands of Guadeloupe and Martinico, they found that the
length of an isochronal pendulum in those islands was 3 feet and 6^ lines.
After this, M. Couplet, the son, in the month of July 1697, at the Royal
Observatory of Paris, so fitted his pendulum clock to the mean motion of
the sun, that for a considerable time together the clock agreed with the
motion of the sun. In November following, upon his arrival at Lisbon, he
found his clock to go slower than before at the rate of 2 13" in 24 hours.
And next March coming to Paraiba, he found his clock to go slower than
at Paris, and at the rate 4 12" in 24 hours ; and he affirms, that the pen
dulum vibrating in seconds was shorter at Lisbon by 2 lines, and at Pa
raiba, by 3 1 lines, than at Paris. He had done better to have reckoned
those differences \\ and 2f : for these differences correspond to the differ
ences of the times 2 13" and 4 12". But this gentleman s observations
are so gross, that we cannot confide in them.
In the following years, 1699, and 1700, M. des Hayes, making another
