there would arise equal floods, which, meeting with as many equal ebbs,
would so balance one the other, that, for that day, the water wrould stag
nate, and remain quiet. If the moon then declined from the equator, the
tides in the ocean would be alternately greater and less, as was said; and
from hence two greater and two lesser tides would be alternately propa
gated towards that port. But the two greater floods would make the
greatest height of the waters to fall out in the middle time betwixt both,
and the &Greater and lesser floods would make the waters to rise to a mean
height in the middle time between them; and in the middle time between
540 THE SYSTEM OF THE WORLD.
the two lesser floods the waters would rise to their least height. Thus in
the space of twenty-four hours the waters would come, riot twice, but once
only to their greatest, and once only to their least height ; and their great
est height, if the moon declined towards the elevated pole, would happen
at the sixth or thirtieth hour after the appulse of the moon to the meridian
and when the moon changed its declination, this flood would be changed
into an ebb.
Of all which we have an example in the port of Batsham, in the king
dom of Tunquin. in the latitude of 20 50 north. In that port, on the
day which follows after the passage of the moon over the equator, the
waters stagnate ; when the moon declines to the north, they begin to fluw
and ebb, not twice, as in other ports, but once only every day ; and the
flood happens at the setting, and the greatest ebb at the rising of the moon.
This tide increaseth with the declination of the moon till the seventh or
eighth day ; then for the seventh or eighth day following it decreaseth at
the same rate as it had increased before, and ceaseth when the moon
changeth its declination. After which the flood is immediately changed
into an ebb ; and thenceforth the ebb happens at the setting and the flood
at the rising of the moon, till the moon again changes its declination.
There are two inlets from the ocean to this port; one more direct and short
between the island Hainan and the coast of QuanttiHg, a province of
China ; the other round about between the same island and the coast of
Cochim ; and through the shorter passage the tide is sooner propagated to
Batsham.
In the channels of rivers the influx and reflux depends upon the current
of the rivers, which obstructs the ingress of the waters from the sea. and
promotes their egress to the sea, making the ingress later and slower, and
the egress sooner arid faster; and hence it is that the reflux is of longer
duration that the influx, especially far up the rivers, where the force of the
sea is less. So Sturmy tells us, that in the river Avon, three miles below
Bristol, the water flows only five hours, but ebbs seven ; and without doubt
the difference is yet greater above Bristol, as at Carcs/iam or the Bath.
This difference does likewise depend upon the quantity of the flux and re
flux
;
for the more vehement motion of the sea near the syzygies of the
luminaries more easily overcoming the resistance of the rivers, will make
the ingress of the water to happen sooner and to continue longer, and will
therefore diminish this difference. But while the moon is approaching to
the syzygies, the rivers will be more plentifully filled, their currents being
obstructed by the greatness of the tides, and therefore will something more
retard the reflux of the sea a little after than a little before the syzygies.
Upon which account the slowest tides of all will not happen in the syzy-
^ies, but precede them a little
; and I observed above that the tides before
the sy/ygies were also retarded by the force of the sun ; and from both
THE SYSTEM OF THE WORLD. 541
causes conjoined the retardation of the tides will be both greater and sooner
before the syzygies. All which I find to be so, by the tide-tables which
Ftamsted has composed from a great many observations.
By the laws we have been describing, the times of the tides are governed ;
but the greatness of th-e tides depends upon the greatness of the seas. Let
C represent the centre of the earth, EAUB the oval figure of the seas, CA
the longer semi-axis of this oval, OB the shorter insisting at right angles
upon the former, D the middle point between A and B, and EOF or eCf
the angle at the centre of the earth, subtended by the breadth of the sea
that terminates in the shores E, F, or e,f. Now, supposing that the point
A is in the middle between the points E, F, and the point D in the middle
between the points e,/, if the difference of the heights CA, CB, represent
the quantity of the tide in a very deep sea surrounding the whole earth,
the excess of the height CA above the height OE or OF will represent the
quantity of the tide in the middle of the sea EF, terminated by the shores
E, F ; and the excess of the height Ce above the height Cf will nearly
represent the quantity of the tide on the shores/" of the same sea. Whence
it appears that the tides are far less in the middle of the sea than at the
shores ; and that the tides at the shores are nearly as EF (p. 451, 452), the
breadth of the sea not exceeding a quadrantal arc. And hence it is that
near the equator, where the sea between Africa and America is narrow,
the tides are far less than towards either side in the temperate zones, wrhere
the seas are extended wider ; or on almost all the shores of the Pacific sea;
as well towards America as towards China,, and within as well as without
the tropics ; and that in islands in the middle of the sea they scarcely rise
higher than two or three feet, but on the shores of great continents are
three or four times greater, and above, especially if the motions propagated
from the ocean are by degrees contracted into a narrow space, and the water,
to fill and empty the bays alternately, is forced to flow and ebb with great
violence through shallow places ; as Plymouth and Chepstow Bridge in
England) at the mount of >S*/. Michael and town of Avranches in Aormcihdy,
and at Cambaia and Peyn. in the East Indies. In which places.
642 THE SYSTEM OF THE WORLD.
the sea, hurried in and out with great violence, sometimes lays the shores
under water, sometimes leaves them dry, for many miles. Nor is the force
of the influx and efflux to be broke till it has raised or depressed the water
to forty or fifty feet and more. Thus also -long and shallow straits that
open to the sea with mouths wider and deeper than the rest of their chan
nel (such as those about Britain and the Magellanic Straits at the east
ern entry) will have a greater flood and ebb, or will more intend and remit
their course, and therefore will rise higher and be depressed lower. Or
the coast of South America it is said that the Pacific sea in its reflux
sometimes retreats two miles, and gets out of sight of those that stand on
shore. Whence in these places the floods will be also higher ; but in deepei
waters the velocity of influx and efflux is always less, and therefore tlu
ascent and descent is so too. Nor in such places is the ocean known to
ascend to more than six, eight, or ten feet. The quantity of the ascent I
compute in the following manner
Let S represent the sun, T the
earth (419. 420), P the moon,
PAGB the moon s orbit. In SP
take SK equal to ST and SL to
SK in the duplicate ratio of SK
to SP. Parallel to PT draw LM ;
and, supposing the mean quantity
of the circum-solar force directed towards the earth to be represented \j
the distance ST or SK, SL will represent the quantity thereof directed
towards the moon. But that force is compounded of the parts SM, LM ;
of which the force LM and that part of SM which is represented by TJVI,
do disturb the motion of the moon (as appears from Prop. LXVI, and its
Corollaries) In so far as the earth and moon are revolved about their
common centre of gravity, the earth will be liable to the action of the like
forces. But we may refer the sums as well of the forces as of the motions
to the moon, and represent the sums of the forces by the lines TM and
ML, which are proportional to them. The force LM, in its mean quan
tity, is to the force by which the moon may be revolved in an orbit, about
the earth quiescent, at the distance PT in the duplicate ratio of the moon s
periodic time about the earth to the earth s periodic time about the nun
(by Cor. XVII, Prop. LXVI) : that is, in the duplicate ratio of 27d
. 7h
.
43 to 365d
. 6h
. 9 ;
or as 1000 to 178725, or 1 to 178f f. The force by
which the moon may be revolved in its orb about the earth in rest, at the
distance PT of 60| semi-diameters of the earth, is to the force by which
it may revolve in the same time at the distance of 60 semi- diameters as
60i to 60 ; and this force is to the force of gravity with us as 1 to 60 X
60 nearly ; and therefore the mean force ML is to the force of gravity at
the surface of the earth as 1 X 60| to 60 X 60 X 178 f, or 1 to
THE SYSTEM OF THF. WORLD. 543
638092,6. Whence the force TM will be ulso given from the proportion
of the lines TM, Ml,. And these are the forces of the sun, by which the
moon s motions are disturbed.
If from the moon s orbit (p. 449V we descend to the earth s surface, those
forces will be diminished in the ratio of the distances 60| and 1 ; and
therefore the force LM will then become 3S604600 times less than the
force of gravity. But this force acting equally every where upon the
earth, will scarcely effect any change on the motion of the sea, and there
fore may be neglected in the explication of that motion. The other force
I M, in places where the sun is vertical, or in their nadir, is triple the
quantity of the force ML, and therefore but 12868200 times less than the
force of gravity.
Suppose now AUBE to represent the spherical surface of the enrth,
</D/>E the surface of the water overspreading it, C the centre of both, A
the place to winch the sun is vertical, B the place opposite : I), E. places
at 90 degrees distance from the former ; ACEwz/A a right angled cylmdric
canal passing through the earth s centre. The force TM in any place is
as the distance of the place from the plane DE, on which a line fr^m A
to C insists at right angles, and J)
therefore in the part of the ca
nal which is represented by EC
ini is of no quantity, but in the
other part AClk is as the gravity
at the several heights ;
for in /
descending towards the centre of 7; -pi
the earth, gravity is (by Prop- ;
LXX1II) every where as the
height ;
and therefore the force
TM drawing the water upwards
will diminish its gravity in the
leg AC//J of the canal in a given
ratio : upon which account the
water will ascend in this leg, till its defect of gravity is supplied by its
greater height : nor will it rest in an equilibrium till its total gravity
becomes equal to the total gravity in EC/m, the other leg of the canal.
Because the gravity of every particle is as its distance from the earth s
centre, the weight of the whole water in either leg will increase in the
duplicate ratio of the height ; and therefore the height of the water in the
leg AC/A* will be to the height thereof in the leg C/wE in the subduplicate
ratio of the number 12868201 to 12808200, or in the ratio of the
number 25623053 to the number 25623052, and the height of the water
in the leg EC/ra to the difference of the heights, as 25623052 to 1. But
the height in the lea: EC/m is of 19615800 Parift feet, as hits been lately
544 THE SYSTEM OF THE WORLD.
found by the mensuration of the French ; and, therefore, by the preceding
analogy, the difference of the heights comes out 9} inches of the Paris
foot
; and the sun s force will make the height of the sea at A to exceed
the height of the same at E by 9 inches. And though the water of the
canal ACE/??7/,: be supposed to be frozen into a hard and solid consistence,
yet the heights thereof at A and E, and all other intermediate places, would
still remain the same.
Let Act (in the following figure) represent that excess of height of 9
inches at A, and hf the excess of height at any other place h; and upon
DC let fall the perpendicular /G, meeting the globe of the earth in F :
and because the distance of the sun ib so great that all the right lines
drawn thereto may be considered as parallel, the force TM in any place /
will be to the same force in the place A as the sine FG to the radius AC.
And, therefore, since those forces tend to the sun in the direction of par
allel lines, they will generate
the parallel heights F/ An,
in the same ratio ; and there
fore the figure of the water
Ylfaeb will be a spheroid
made by the revolution of an
ellipsis about its longer axis
ab. And the perpendicular
height fh will be to the ob
lique height F/ as/G to /C,
or as FG to AC : and there
fore the height fh is to the
height Art in the duplicate
ratio of FG to AC, that is, in the ratio of the versed sine of double the
angle DC/ to double the radius, and is thence given. And hence to the
several moments of the apparent revolution of the sun about the earth we
may infer the proportion of the ascent and descent of the waters at any
given place under the equator, as well as of the diminution of that ascent
and descent, whether arising from the latitude of places or from the sun s
declination ; viz., that on account of the latitude of places,
the ascent and
descent of the sea is in all places diminished in the duplicate ratio of the
co-sines of latitude ; and on account of the sun s declination, the ascent
and descent under the equator is diminished in the duplicate ratio of the
v)-sine of declination. And in places without the equator the half sum
of the morning and evening ascents (that is, the mean ascent) is diminished
nearly in the same ratio.
Let S and L respectively represent the forces of the sun and moon
placed in the equator, and at their mean distances from the earth; R the
radius ; T and V the versed sines of double the complements of the sun
THE SYSTEM OJ THE WORLD. 545
and moon s declinations to any given time ; D and E the moan apparent
diameters of the sun and moon : and, supposing F and G to be their appa
rent diameters to that given time, their forces to raise the tides under the
VG 3 TF 3
equator will be, in the
syzygies-^ ^ 1, -f
^ 3 S; in the quadratures,
VG 3 TF 3
--, L -TTT S. And if the same ratio is likewise observed under 2RE 3 2R1) 3
the parallels, from observations accurately made in our northern climates
we may determine the proportion of the forces L and S ; and then by
means of this rule predict the quantities of the tides to every syzygy and
quadrature.
At the mouth of the river Avon, three miles below Bristol (p. 450 to
453), in spring and autumn, the whole ascent of the water in the conjunc
tion or opposition of the luminaries (by the observation of Sturnty) is
about 45 feet, but in the quadratures only 25. Because the apparent di
ameters of the luminaries are not here determined, let us assume them in
their mean quantities, as well as the moon s declination in the equinoctial
quadratures in its mean quantity, that is, 23| ; and the versed sine of
double its complement will be 1082, supposing the radius to be 1000. But
the declinations of the sun in the equinoxes and of the moon in the syzygies
are of no quantity, and the versed sines of double the complements
are each 2000. Whence those forces become L + S in the syzygies, and
\ L S in the quadrature^ respectively proportional to the heights
/cUOU
of the tides of 45 and 25 feet, or of and 5 paces. And, therefore, mul-
15138
tiplying the extremes and the means, we have 5L + 5S = TxTr L<
But farther
;
I remember to have been told that in summer the ascent of
the sea in the syzygies is to the ascent thereof in the quadratures as about
5 to 4. In the solstices themselves it is probable that the proportion may
be something less, as about 6 to 5 ; whence it would follow that L is =
5|S [for then the proportion is L + S : I, -S : : 6 : 5].
Till we can more certainly determine the proportion from observation, let
us assume L = 5^S ; and since the heights of the tides are as the forces
which excite them, and the force of the sun is able to raise the tides to the
height of nine inches, the moon s force will be sufficient to raise the same
to the height of four feet. And if we allow that this height may be
doubled, or perhaps tripled, by that force of reciprocation which we observe
in the motion of the waters, and by which their motion once be ^un is kept
35
546 THE SYSTEM OF THE WORLD.
up for some time, there will be force enough to generate all that quantity
of tides which we really find in the ocean.
Thus we have seen that these forces are sufficient to move the sea. But.
so far as I can observe, they will not be able to produce any other effect
sensible on our earth ; for since the weight of one grain in 4000 is not
sensible in the nicest balance : and the sun s force to move the tides is
12868200 less than the force of gravity ;
arid the sum of the forces of both
moon and sun, exceeding the sun s force only in the ratio of 6^ to 1, is still
2032890 times less than the force of gravity ;
it is evident that both forces
together are 500 times less than what is required sensibly to increase * r
diminish the weight of any body in a balance. And, therefore, they will
not sensibly move any suspended body ; nor will they produce any sensible
eifect on pendulums, barometers, bodies swimming in stagnant water, or in
the like statical experiments. In the atmosphere, indeed, they will excite
such a flux and reflux as they do in the sea, but with so small a motion
that no sensible wind will be thence produced.
if the effects of both moon and sun in raising the tides (p. 454), as well
as their apparent diameters, were equal among themselves, their absolute
forces would (by Cor. XIV, Prop. LXVI) be as their magnitudes. But the
effect of the moon is to the effect of the sun as about 5| to 1
; and the
moon s diameter less than the sun s in the ratio of 31 1 to 32^, or of 45 to
46. Now the force of the moon is to be increased in the ratio of the effect
directly, and in the triplicate ratio of the diameter inversely. Whence the
force of the moon compared with its magnitude will be to the force of the
sun compared with its magnitude in the ratio compounded of 5-^- to 1, and
the triplicate of 45 to 46 inversely, that is, in the ratio of about 5^ to 1.
And therefore the moon, in respect of the magnitude of its body, has an
absolute centripetal force greater than the sun in respect of the magnitude
of its body in the ratio to 5 T\ to 1, and is therefore more dense in the
same ratio.
In the time of 27 1
. 7h
. 43 , in which the moon makes its revolution about
the earth, a planet may be revolved about the sun at the distance of 18.95 1
diameters of the sun from the sun s centre, supposing the mean apparen
diameter of the sun to be 32} ; and in the same time the moon may be r"-
volved about the earth at rest, at the distance of 30 of the earth s diame
ters. If in both cases the number of diameters was the same, the absolute
circum-terrestrial force would (by Cor. II, Prop. LXXll) be to the absolute
circum-solar force as the magnitude of the earth to the magnitude of the
tun. Because the number of the earth s diameters is greater in the ratio
of 30 to 18,954, the body of the earth will be less in the triplicate of that
ratio, that is, in the ratio of 3|| to 1. Wherefore the earth s force, for the
magnitude of its body, is to the sun s force, for the magnitude of its body,
as 3f f to 1 : and consequently the earth s density to the sun s will be IL
THE SYSTEM OF THE WORLD 547
the same ratio. Since, then, the moon s density is to the sun s density as
5JS to I, the moon s density will be to the earth s density as 5 r\ to 3f {,
or as 23 to 16. Wh. veforc since the moon s magnitude is to the earth s
magnitude as about I to 4l, the moon s absolute centripetal force will be
to the earth s absolute centripetal force as about I to 29, and the quantity
of matter in the moon to the quantity of matter in the earth in the sameratio.
And hence the common centre of gravity of the earth and moon is
more exactly determined than hitherto has been done; from the knowledge
of which AVC may now infer the moon s distance from the earth with greater
accuracy. But I would rather wait till the proportion of the bodies of the
moon and earth one to the other is more exactly defined from the phae
nomena of the tides, hoping that in the mean time the circumference of the
earth may be measured from more distant stations than any body has yet
employed for this purpose.
Thus I have given an account of the system of the planets. As to the
fixed stars, the smallness of their annual parallax proves them to be re
moved to immense distances from the system of the planets: that this
parallax is less than one minute is most certain
; and from thence it follows
that the distance of the fixed stars is above 360 times greater than the
distance of Saturn from ;he sun. Such as reckon the earth one of the
planets, and the sun one of the fixed stars, may remove the fixed stars to
yet greater distances by the following arguments: from the annual motion
of the earth there would happen an apparent transposition of the fixed
stars, one in respect of another, almost equal to their double parallax: but
the greater and nearer stars, in respect of the more remote, which are only
seen by the telescope, have not hitherto been observed to have the least
motion. If we should suppose that motion to be but less than 20", the
distance of the nearer fixed stars would exceed the mean distance of Saturn
by above 2000 times. Again: the disk of Saturn, which is only 17" or
18" in diameter, receives but about ^---- --^.^ of the sun s light; for so
much less is that disk than the whole spherical surface of the orb of Saturn.
Now if we suppose Saturn to rellec* about { of this light, the whole light
reflected from its illuminated hemisphere will be about T ^^Wo o"^~ ^ ^e
whole light emitted from the sun s hemisphere: and, therefore, since light
is rarefied in the duplicate ratio of the distance from the luminous body, if
the sun was 10000 v/42 times more distant than Saturn, it would yet ap
pear as lucid as Saturn now does without its ring, that is, something more
lucid than a fixed star of the first magnitude. Let us, therefore, suppose
that the distance from which the sun would shine as a fixed star exceeds
that of Saturn by about 100,000 times, and its apparent diameter will be
7V
. 16vi
. and its parallax arising from the annual motion of the earth 13"" :
and so great will be the distance, the apparent diameter, and the parallax
of the fixed stars of the first magnitude, in bulk and light equal to our sun.
4>i THE SYSTEM OF THE WORLD.
Some may, perhaps, imagine that a great part of the light of the fixed stars
is intercepted and lost in its passage through so vast spaces, and upon that
account pretend to place the fixed stars at nearer distances; but at this
rate the remoter stars could be scarcely seen. Suppose, for example, that
of the light perish in its passage from the nearest fixed stars to us ; then
| will twice perish in its passage through a double space, thrice through a
triple, and so forth. And, therefore, the fixed stars that are at a double
distance wHl be 16 times more obscure, viz., 4 times more obscure on ac
count of the diminished apparent diameter ; and, again, 4 times more on
account of the lost light. And, by the same argument, the fixed stars at a
triple distance will be 9 X 4. X 4, or 144 times more obscure; and those
at a quadruple distance will be 16 X 4 X 4 X 4, or 1024 times more ob
scure: but so great a diminution of light is no ways consistent with the
phenomena and with that hypothesis which places the fixed stars at differ
ent distances.
Tne fixed stars being, therefore, at such vast distances from one another
(p. 460, 461), can neither attract each other sensibly, nor be attracted by
our sun. But the comets must unavoidably be acted on by the circum
solar force ; for as the comets were placed by astronomers above the moon,
