orbits (p 403), and, by radii drawn to the sun, describe areas nearly pro
portional to the times, as is explained in Prop. LXV. If the sun was qui
escent, and the other planets did not act mutually one upon another, their
orbits would be elliptic, and the areas exactly proportional to the times (by
Prop. XI, and Cor. 1, Prop. XIII). But the actions of the planets amonir
themselves, compared with the actions of the sun on the planets, are of no
moment, and produce no sensible errors. And those errors are less in rev
olutions about the sun agitated in the manner but now described than if
those revolutions were made about the sun quiescent (by Prop. LXV1, and
Cor. Prop. LXVIll), especially if the focus of every orbit is placed in the
common centre of gravity of all the lower included planets; viz., the focus
of the orbit of Mercury in the centre of the sun : the focus of the orbit of
Venus in the common centre of gravity of Mercury and the sun ; the focus
of the orbit of thp earth in the common centre of gravity of Venus, Mer
cury, and the sun ; and so of the rest. And by this means the foci of the
crbits of all the planets, except Saturn, will not be sensibly removed from
the centre of the sun, nor will the focus of the orbit of Saturn recede sensi
bly from the common centre of gravity of Jupiter and the sun. And
therefore astronomers are not far from the truth, when they reckon the
sun s centre the common focus of all the planetary orbits. In Saturn itself
THE SYSTEM CF THE W )RLD.
the error thence arising docs not exceed 1 45 . And if its orbit, by placing
the focus thereof in the common centre of gravity of Jupiter and the sun,
ghall happen to agree better with the phenomena, from thence all that we
have said will be farther confirmed.
If the sun was quiescent, and the planets did not act one on another, the
aphelions and nodes of their orbits would likewise (by Prop. 1, XI, and Cor.
Prop. XIU) be quiescent. And the longer axes of their elliptic orbits
would (by Prop. XV) be as the cubic roots of the squares of their periodic
times : and therefore from the given periodic times would be also given.
But those times are to be measured not from the equinoctial points, which
are rnoveable, but from the first star of Aries. Put the semi-axis of the
earth s orbit 100000, and the semi-axes of the orbits of Saturn, Jupiter,
Mars, Venus, and Mercury, from their periodic times, will come out
953806, 520116, 152399, 72333, 38710 respectively. But from the sun s
motion every semi-axis is increased (bv Prop. LX) by about one third of
the distance of the sun s centre from the common centre of gravity of
the sun and planet (p. 405, 406.) And from the actions of the exterior
planets on the interior, the periodic times of the interior are something
protracted, though scarcely by any sensible quantity ; and their aphelions
are transferred (by Cor. VI and VII, Prop. LXVI)by very slow motions
in conset/ue/ttia. And on the like account the periodic times of all, espe
cially of the exterior planets, will be prolonged by the actions of the
somets, if any such there are, without the orb of Saturn, and the aphe
lions of all will be thereby carried forwards in consequent-la. But from
the progress of the aphelions the regress of the nodes follows (by Cor.
XI, XIII, Prop. 1 jXVI). And if the plane of the ecliptic is quiescent, the
regress of the nodes (by Cor. XVI, Prop. LX.VI) will be to the progress of
*he aphelion in every orbit as the regress of the nodes of the moon s orbit
to the progress of its apogeon nearly, that is, as about 10 to 21. But as
tronomical observations seem to confirm a very slow progress of the aphe
lions, and a regress of the nodes in respect of the fixed stars. And hence
it is probable that there are comets in the regions beyond the planets, which,
revolving in very eccentric orbs, quickly fly through their perihelion parts,
and, by an exceedingly slow motion in their aphelions, spend almost their
whole time in the regions beyond the planets ;
as we shall afterwards ex
plain more at large.
The planets thus revolved about the sun (p. 413, 41.4, 415) may at the
same time carry others revolving about themselves as satellites or moons,
as appears by Prop. LXVI. But from the action of the sun our moon
must move with greater velocity, and, by a radius drawn to the earth, de
scribe an area greater for the time ;
it must have its orbit less curve, and
therefore approach nearer to the earth in the syzygies than in the quadra
tures, except in so far as the motion of eccentricity hinders those effects.
THE SYSTEM OF THE WORLD. 533
Per the eccentricity is greatest when the moon s apogeon is in the syzygies,
and least when the same is in the quadratures ; and hence it is that the
perigeon moon is swifter and nearer to us, but the apogeon moon slower and
farther from us, in the syzygies than in the quadratures. But farther; the
apogeon has a progressive and the nodes a regressive motion, both unequa
ble. For the apogeon is more swiftly progressive in its syzygies, more
slowly regressive in its quadratures, and by the excess of its progress above
its regress is yearly transferred in coiisequentia ; but the nodes are quies
cent in their syzygies, and most swiftly regressive in their quadratures. But
farther, still, the greatest latitude of the moon is greater in its quadra
tures than in its syzygies ; and the mean motion swifter in the aphelion of
the earth than in its perihelion. More inequalities in the moon s motion
have not hitherto been taken notice of by astronomers : but all these fol
low from our principles in Cor. II, III, IV, V, VI, VII, VIII, IX, X, XI,
XII, XIII, Prop. LXVI, and are known really to exist in the heavens.
And this may seen in that most ingenious, and if I mistake not, of all, the
most acccurate, hypothesis of Mr. Horrnx, which Mr. Flamsted has fitted
to the heavens ; but the astronomical hypotheses are to be corrected in the
motion of the nodes ;
for the nodes admit the greatest equation or prosthaphaeresis
in their octants, and this inequality is most conspicuous when
the moon is in the nodes, and therefore also in the octants ; and hence it
was that Tycho, and others after him, referred this inequality to the
octants of the moon, and made it menstrual; but the reasons by us addu
ced prove that it ought to be referred to the octants of the nodes, and to
be made annual.
Beside those inequalities taken notice of by astronomers (p. 414, 445,
447,) there are yet some others, by which the moon s motions are so dis
turbed, that hitherto by no law could they be reduced to any certain regu
lation. For the velocities or motions of the apogee and nodes of
the moon, and their equations, as well as the differs ice betwixt the greatest
eccentricity in the syzygies and the least in the < rrdratures, and that ine
quality which we call the variation, in the progress of the year are aug
mented and diminished (by Cor. XIV, Prop. LXVI) in the triplicate ratio
of the sun s apparent diameter. Beside that, the variation is mutable
rly in the duplicate ratio of the time between the quadratures (by Cor.
I and II, Lem. X, and Cor. XVI, Prop. LXVI); and all those inequali
ties are something greater in that part of the orbit which respects the sun
than in the opposite part, but by a difference that is scarcely or not at all
perceptible.
By a computation (p. 422), which for brevity s sake I do not describe, 1
also find that the area which the moon by a radius drawn to the earth
describes in the several equal moments of time is nearly as the sum of the
number 237T\, and versed sine of the double distance of the moon frour.
534 THE SYSTEM OF THE WORLD.
the nearest quadrature in a circle whose radius is unity ; and therefore
that the square of the moon s distance from the earth is as that sum divid
ed by the horary motion of the moon. Thus it is when the variation in
the octants is in its mean quantity ; but if the variation is greater or less,
that versed sine must be augmented or diminished in the same ratio. Let
astronomers try how exactly the distances thus found will agree with fjie
moon s apparent diameters.
From the motions of our moon we may derive the motions of themoon*
or satellites of Jupiter and Saturn (p. 413); for the mean motion of the
nodes of the outmost satellite of Jupiter is to the mean motion of the nodes
of our moon in a proportion compounded of the duplicate proportion of
the periodic time of the earth about the sun to the periodic time of Jupiter
about the sun, and the simple proportion of the periodic time of the sat
ellite about Jupiter to the periodic time of our moon about the earth (by
Gor. XVI, Prop. LXVI) : and therefore those nodes, in the space of a hun
dred years, are carried 8 24 backwards, or in atitecedeutia. The mean
motions of the nodes of the inner satellites are to the (mean) motion of
(the nodes of) the outmost as their periodic times to the periodic time of
this, by the same corollary, and are thence given. And the motion of the
apsis of every satellite in consequentia is to the motion of its nodes in
a/ttecedentia, as the motion of the apogee of our moon to the motion of i s
nodes (by the same Corollary), and is thence given. The greatest equa
tions of the nodes and line of the apses of each satellite are to the greatest
equations of the nodes and the line of the apses of the moon respectively
as the motion of the nodes and line of the apses of the satellites in the
time of one resolution of the first equations to the motion of the nodes
and apogeon of the moon in the time of one revolution of the last equa
tions. The variation of a satellite seen from Jupiter is to the variation
of our moon in the same proportion as the whole motions of their nodes
respectively, during the times in which the satellite and our moon (after
parting from) arc revolved (again) to the sun, by the same Corollary ; ami
therefore in the outmost satellite the variation does not exceed 5" 12 ".
From the small quantity of those inequalities, and the slowness of the
motions, it happens that the motions of the satellites are found to be so
regular, that the more modern astronomers either deny all motion to the
nodes, or affirm them to be very slowly regressive.
(P. 404). While the planets are thus revolved in orbits about remote
centres, in the mean time they make their several rotations about their
proper axes; the sun in 26 days; Jupiter in 9h
. 56 ; Mars in 24f,
h
.
;
Venus in 23h
.
; and that in planes not much inclined to the plane of the
ecliptic, and according to the order of the signs, as astronomers determine
from the spots or macula? that by turns present themselves to our sight in
their bodies; and there is a like revolution of our earth performed in 24h
.
;
THE SYSTEM OF THE WORLU. 535
find those motions are neither accelerated nor retarded
l>y
the actions of
the centripetal forces, as appears by Cor. XXII, Prop. LXVI ; and there
fore of all others they are the most equable and most fit for the mensura
tion of time; but those revolutions are to be reckoned equable not from
their return to the sun, but to some fixed star: for as the position of the
planets to the sun is unequably varied, the revolutions of those planets
from sun to sun are rendered unequable.
In like manner is the moon revolved about its axis by a motion most
equable in respect of the fixed stars, viz., in 27 J
. 7h
. 43 , that is, in the
space of a sidereal month ;
so that this diurnal motion is equal to the
mean motion of the moon in its orbit : upon which account the same face
of the moon always respects the centre about which this mean motion is
performed, that is, the exterior focus of the moon s orbit nearly ; and hence
arises a deflection of the moon s face from the earth, sometimes towards
the east, and other times towards the west, according to the position of the
focus which it respects ; and this deflection is equal to the equation of the
moon s orbit, or to the difference betwixt its mean and true motions; and
this is the moon s libration in longitude: but it is likewise affected with
a libration in latitude arising from the inclination of the moon s axis to
the plane of the orbit in which the moon is revolved about the earth
;
for
that axis retains the same position to the fixed stars nearly, and hence the
poles present themselves to our view by turns, as we may understand from
the example of the motion of the earth, whose poles, by reason of the incli
nation of its axis to the plane of the ecliptic, are by turns illuminated by
the sun. To determine exactly the position of the moon s axis to the
fixed stars, and the variation of this position, is a problem worthy of an
astronomer.
By reason of the diurnal revolutions of the planets, the matter which
they contain endeavours to recede from the axis of this motion ; and hence
the fluid parts rising higher towards the equator than about the poles
(p. 405), would lay the solid parts about the equator under water, if those
parts did not rise also (p. 405, 409) : upon which account the planets are
something thicker about the equator than about the poles ; and their equi
noctial points (p. 413) thence become regressive ; and their axes, by a
motion of nutation, twice in every revolution, librate towards their eclip
tics, and twice return again to their former inclination, as is explained in
Cor. XVIII, Prop. LXVI ; and hence it is that Jupiter, viewed through
very long telescopes, does not appear altogether round (p. 409). but having
its diameter that lies parallel to the ecliptic something longer than that
which is drawn from north to south.
And from the diurnal motion and the attractions (p. 415, 418) of the
Bun and moon our sea ought twice to rise and twice to fall every day, as
well lunar as solar (by Cor. XIX, XX, Prop. LXVI), and the greatest
636 THE SYSTEM OF THE WORLD.
height of the water to happen before the sixth hour of either day and aftei
the twelfth hour preceding. By the slowness of the diurnal motion the
flood is retracted to the twelfth hour ; and by the force of the motion of
reciprocation it is protracted and deferred till a time nearer to the sixth
hour. But till that time is more certainly determined by the pheno
mena, choosing the middle between those extremes, why may we not
conjecture the greatest height of the water to happen at the third hour ?
for thus the water will rise all that time in which the force of the lumi
naries to raise it is greater, and will fall all that time in which their force
is less : viz., from the ninth to the third hour when that force is greater,
and from the third to the ninth when it is less. The hours I reckon from
the appulse of each luminary to the meridian of the place, as well under
as above the horizon ; and by the hours of the lunar day I understand the
twenty-fourth parts of that time which the moon spends before it comes
about again by its apparent diurnal motion to the meridian of the place
which it left the day before.
But the two motions which the two luminaries raise will not appear distin
guished, but will make a certain mixed motion. In the conjunction or op
position of the luminaries their forces will be conjoined, and bring on the
greatest flood and ebb. In the quadratures the sun will raise the waters
which the moon dcpresseth. and depress the waters which the moon raiseth
;
and from the difference of their forces the smallest of all tides will follow.
And because (as experience tells us) the force of the moon is greater than
that of the sun, the greatest height of the water will happen about the
third lunar hour. Out of the syzygies and quadratures the greatest tide
which by the single force of the moon ought to fall out at the third lunar
hour, and by the single force of the sun at the third solar hour, by the
compounded forces of both must fall out in an intermediate time that ap
proaches nearer to the third hour of the moon than to that of the sun:
and, therefore, while the moon is passing from the syzygies to the quadra
tures, during which time the third hour of the sun precedes the third of
the moon, the greatest tide will precede the third lunar hour, and that by
the greatest interval a little after the octants of the moon ; and by like
intervals the greatest tide will follow the third lunar hour, while the moon
is passing from the quadratures to the syzygies.
But the effects of the luminaries depend upon their distances from the
earth ; for when they are less distant their effects are greater, and when
more distant their effects are less, and that in the triplicate proportion of
their apparent diameters. Therefore it is that the sun in the winter time,
being then in its perigee, has a greater effect, and makes the tides in the
syzyii ies something greater, and those in the quadratures something less,
cre/m.<? panbiis, than in the summer season ; and every month the moon,
vhile in the perigee, raiseth greater tides than at the distance of 15 days
THE SYSTEM OF THE WORLD. 537
K N
K forc or after, when it is in its apogee. Whence it comes to pasa that two
nighest tides do not follow one the other in two immediately succeeding
syzygies.
The effect of either luminary doth likewise depend upon its declination
or distance from the equator ;
for if the luminary was placed at the pole,
it would constantly attract all the parts of the waters, without any inten
sion or remission of its action, and could cause no reciprocation of motion ;
and, therefore, as the luminaries decline from the equator towards either
pole, they will by degrees lose their force, and on this account will excite
lesser tides in the solstitial than in the equinoctial syzygies. But in the
solstitial quadratures they will raise greater tides than in the quadratures
about the equinoxes ; because the effect of the moon, then situated in the
equator, most exceeds the effect of the sun ; therefore the greatest tides
fall out in those syzygies. and the least in those quadratures, which happen
about the time of both equinoxes ; and the greatest tide in the syzygies is
always succeeded by the least tide in the quadratures, as we lind by expe
rience. But because the sun is less distant from the earth in winter than
in summer, it cornes to pass that the greatest and least tides more fre
quently appear before than after the vernal equinox, and more frequently
after than before the autumnal.
Moreover, the effects cf che lumina
ries depend upon the latitudes of places.
Let AjoEP represent the earth on all
sides covered with deep waters: C its
centre; P, p, its poles; AE the equa
tor: P any place without the equator:
F/ the parallel of the place : Del the
correspondent parallel OD the other side
of the equator; L the rlnoe which the moon possessed three hours before
H the place of the earth directly under it
; h the opposite place ; K, k,
the places at 90 degrees distance ; CH, Ch, the greatest heights of the sea
from the centre of the earth ; and CK, C&, the least heights : and if with
the axes H/?,, K/r, an ellipsis is described, and by the revolution rf that
ellipsis about its longer axis HA a spheroid HPK//jt?A* is formed, this sphe
roid will nearly represent the figure of the sea; and CF, C/, CD, Cd, will
represent the sea in the places F,/, D, d. But farther : if in the said revo
lution of the ellipsis any point N describes the circle NM, cutting the
parallels F/, Dr/?
in any places R, T, and the equator AE in S, CN will
represent the height of the sea in all those places R, S, T, situated in this
circle. Wherefore in the diurnal revolution of any place F the greatest
flood will be in F. at the third hour after the appulse of the moon to the
meridian above the horizon ; and afterwards the greatest ebb in Q, at the
third hour after the setting of the moon : and then the greatest flood inf.
538 THE SYSTEM OF THE WORLD.
at the third Lour after the appulse of the rnoon to the meridian under tht
horizon , and. lastly, the greatest ebb in Q. at the third hour after the
rising of the moon; and the latter flood
iny" will be less than the preced
ing flood in F For the whole sea is divided into two huge and hemis
pherical floods, one in the hemisphere KH/rC on the north side, the other
in the opposite hemisphere KH/cC, whicli we may therefore call the north
ern and the southern floods : these floods being always opposite the one to
the other, come by turns to the meridians of all places after the interval
of twelve lunar hours ; and, seeing the northern countries partake more
of the northern flood, and the southern countries more of the southern
flood, thence arise tides alternately greater and less in all places without
the equator in Avhich the luminaries rLe and set. But the greater tide
will happen when the moon declines towards the vertex of the place, about
the third hour after the -appulse of the moon to the meridian above the
horizon ; and when the moon changes its declination, that which was the
greater tide will be changed into a lesser
; and the greatest difference of
the floods will fall out about the times of the solstices, especially if the
ascending node of the moon is about the first of Aries. So the morning
tides in winter exceed those of the evening, and the evening tides exceed
those of the morning in summer ; at Plymouth by the height of one foot,
but at Bristol by the height of 15 inches, according to the observations of
Qvleptvss and Stitrnnj.
But the motions which we have been describing suffer some alteration
from that force of reciprocation which the waters [having once received]
retain a little while by their vis iiisita ; whence it comes to pass that the
tides may continue for some time, though the actions of the luminaries
should cease. This power of retaining the impressed motion lessens the
difference of the alternate tides, and makes those tides which immediately
succeed after the syzygies greater, and those which follow next after the
quadratures less. And hence it is that the alternate tides at l
:
1y month
and Bristol do not differ much more one from the other than by the height
of a foot, or of 15 inches; and that the greatest tides <~>f all at those ports
are not the first but the third after the syzygies.
And. besides, all the motions are retarded in their passage through shal
low channels, so that the greatest tides of all, in some strai s and mouths
of rivers, are the fourth, or even the fifth, after the syzygies.
It may also happen that the greatest tide may be the fourth or fifth
after the syzygies, or fall out yet later, because the motions of the sea are
retarded in passing through shallow places towards the shores: for so the
tide arrives at the western coast of Ireland at the third lunar hour, and an
hour or two after at the ports in the southern coast of the same island
; as
also at the islands Cftssiterides, commonly Sorliti^s ; then successively at
Palrnonth. Plymouth, Portland, the isle of Wight, Winchester, Dover,
THE SYSTEM OF THE WORLD. 539
the mouth of the Thames, arid London Btidgey spending twelve hours in
this passage. But farther; the propagation of the tides may he obstructed
even by the channels of the ocean itself, when they are not of depth enough,
for the flood happens at the third lunar hour in the Canary islands ; and
at all those western coasts that lie towards the Atlantic ocean, as of Ire
land, France, Spain, and all Africa, to the Cape of Good Hope, except
in some shallow places, where it is impeded, and falls out later
; and in the
straits of Gibraltar, where, by reason of a motion propagated from the
Mediterranean sea, it flows sooner. But, passing from those coasts over
the breadth of the ocean to the coasts of America, the flood arrives first at
the most eastern shores of Brazil, about the fourth or fifth lunar hour;
then at the mouth of the river of the Amazons at the sixth hour, but at
the neighbouring islands at the fourth hour ; afterwards at the islands of
Bermudas at the seventh hour, and at port St. An^nstin in Florida at
seven and a half. And therefore the tide is propagated through the ocean
with a slower motion than it should be according to the course of the
moon ; and this retardation is very necessary, that the sea at the same time
may fall between Brazil and New France, and rise at the Canary islands,
and on the coasts of Europe and Africa, and vice versa : for the sea can
not rise in one place but by falling in another. And it is probable that
the Pacific sea is agitated by the same laws : for in the coasts of Chili and
Peru the highest flood is said to happen at the third lunar hour. But
with what velocity it is thence propagated to the eastern coasts of
Japan, the Philippine and other islands adjacent to China, I have not
yet learned.
Farther; it may happen (p. 418) that the tide may be propagated from
the ocean through different channels towards the same port, and may pass
quicker through some channels than through others, in which case the
same tide, divided into two or more succeeding one another, may compound
new motions of different kinds. Let us suppose one tide to be divided into
two equal tides, the former whereof precedes the other by the space of six
hours, and happens at the third or twenty-seventh hour from the appulse
of the moon to the meridian of the port. If the moon at the time of this
appulse to the meridian was in the equator, every six hours alternately
