subducted if in consequential and in the octants, where it is of the
greatest magnitude, it arises to 47" in the mean distance of the sun from
the earth, as I find from the theory of gravity. In other distances of the
sun, this equation, greatest in the octants of the nodes, is reciprocally as
the cube of the sun s distance from the earth ; and therefore in the sun s
perigee it comes to about 49", and in its apogee to about 45".
By the same theory of gravity, the moon s apogee goes forward at the
greatest rate when it is either in conjunction with or in opposition to the
sun, but in its quadratures with the sun it goes backward ; and the ec
centricity comes, in the former case, to its greatest quantity ;
in the latter
to its least, by Cor. 7, 8, and 9, Prop. LXVI, Book 1. And those ine
qualities, by the Corollaries we have named, are very great, and generate
the principal which I call the semi-annual equation of the apogee ; and
this semi-annual equation in its greatest quantity comes to about 12 18 ,
as nearly as I could collect from the phenomena. Our countryman,
HorroXj was the first who advanced the theory of the moon s moving in
an ellipsis about the earth placed in its lower focus. Dr. Halley improved
the notion, by putting the centre of the ellipsis in an epicycle whose cen
BOOK III.] OF NATURAL PHILOSOPHY. 447
tre is uniformly revolved about the earth ; and from the motion in this
epicycle the mentioned inequalities in the progress and regress of the apo
gee, and in the quantity of eccentricity, do arise. Suppose the mean dis
tance of the moon from the earth to be divided into 100000 parts, and
let T represent the earth, and TC the moon s mean eccentricity of 5505
such parts. Produce TC to B, so as CB may be the sine of the greatest
semi-annual equation 12 18 to the radius TC; and the circle BOA de
scribed about the centre C, with the
( interval CB, will be the epicycle
spoken of, in which the centre of the
moon s orbit is placed, and revolved
according to the order of the letters
BDA. Set off the angle BCD equal
to twice the annual argument, or
twice the distance of the sun s true place from the place of the moon s
apogee once equated, and CTD will be the semi-annual equation of the
moon s apogee, and TO the eccentricity of its orbit, tending to the place
of the apogee now twice equated. But, having the moon s mean motion,
the place of its apogee, and its eccentricity, as well as the longer axis of
its orbit 200000, from these data the true place of the moon in its orbit,
together with its distance from the earth, may be determined by the
methods commonly known.
In the perihelion of the earth, where the force of the sun is greatest,
the centre of the moon s orbit moves faster about the centre C than in the
aphelion, and that in the reciprocal triplicate proportion of the sun s dis
tance from the earth. But, because the equation of the sun s centre is
included in the annual argument, the centre of the moon s orbit moves
faster in its epicycle BDA, in the reciprocal duplicate proportion of the
sun s distance from the earth. Therefore, that it may move yet faster in
the reciprocal simple proportion of the distance, suppose that from D, the
centre of the orbit, a right line DE is drawn, tending towards the moon s
apogee once equated, that is, parallel to TC ; and set off the angle EDF
equal to the excess of the aforesaid annual argument above the distance
of the moon s apogee from the sun s perigee in conseqiientia ; or; which
comes to the same thing, take the angle CDF equal to the compleiuent of
the sun s true anomaly to 360 ; and let DF be to DC as twice the eccen
tricity of the orbis magnus to the sun s mean distance from the earth.
and the sun s mean diurnal m:tion from the moon s apogee to the sun s
mean diurnal motion from its own apogee conjunctly, that is, as 33f to
1000, and 52 27" 16 " to 59 8" 10 "
conjunctly, or as 3 to 100; and
imagine the centre of the moon s orbit placed in the point F to be revolved
in an epicycle whose centre is D, and radius DF, while the point D moves
in the circumference of the circle DABD : for by this means the centre of
THE MATHEMATICAL PRINCIPLES [BOOK III
the moon s orbit comes to describe a certain curve line about the centre C
with a velocity which will be almost reciprocally as the cube of the sun s
distance from the earth, as it ought to be.
The calculus of this motion is difficult, but may be rendered more easy
by the following approximation. Assuming, as above, the moon s mean
distance from the earth of 100000 parts, and the eccentricity TC of 5505
Buch parts, the-line CB or CD will be found 1172f, and DF 35} of those
parts : and this line DF at the distance TC subtends the angle at the earth,
which the removal of the centre of the orbit from the place D to the place P generates in the motion of this centre; and double this line DF in a
parallel position, at the distance of the upper focus of the moon s orbit from
the earth, subtends at the earth the same angle as DF did before, which
that removal generates in the motion of this upper focus
; but at the dis
tance of the moon from the earth this double line 2DF at the upper focus,
in a parallel position to the first line DF, subtends an angle at the moon,
which the said removal generates in the motion of the moon, which angle
may be therefore called the second equation of the moon s centre
; and this
equation, in the mean distance of the moon from the earth, is nearly as the
sine of the angle which that line DF contains with the line drawn from
the point F to the moon, and when in its greatest quantity amounts to 2
25". But the angle which the line DF contains with the line drawn from
the point F to the moon is found either by subtracting the angle EDF
from the mean anomaly of the moon, or by adding the distance of the moon
from the sun to the distance of the moon s apogee from the apogee of the
sun ; and as the radius to the sine of the angle thus found, so is 2 25" to
the second equation of the centre: to be added, if the forementioned sum
be less than a semi-circle ;
to be subducted, if greater. And from the moon s
place in its orbit thus corrected, its longitude may be found in the syzygies
of the luminaries.
The atmosphere of the earth to the height of 35 or 40 miles refracts the
sun s light. This refraction
^scatters
and spreads the light over the earth s
shadow ; and the dissipated^ light near the limits of the shadow dilates the
shadow. Upon which account, to the diameter of the shadow, as it cornea
out by the parallax, I add 1 or 1^ minute in lunar eclipses.
But the theory of the moon ought to be examined and proved from the
phenomena, first in the syzygies, then in the quadratures, and last of all
in the octants: and whoever pleases to undertake the work will find it
not amiss to assume the following mean motions of the sun and moon at
the Royal Observatory of Greenwich, to the last day of December at noon,
anno 1700, O.S. viz. The mean motion of the sun Y5> 20 43 40", and of
its apogee s 7 44 30"; the mean motion of the moon ^ 15 21 00";
of its apogee, X 8 20 00"; and of its ascending node Si 27 24 20";
and the difference of meridians betwixt the Observatory at Greenwich and
BOOK III.] OF NATURAL PHILOSOPHY. 449
the Royal Observatory at Paris, Oh
. 9 20 : but the mean motion >f the
inoon and of its apogee are not yet obtained with sufficient accuracy.
PROPOSITION XXXVI. PROBLEM XVII.
Tofind the force of the sun to move the sea.
The sun s force Ml, or PT to disturb the motions of the moon, was (by
Prop. XXV.) in the moon s quadratures, to the force of gravity with us, as
1 to 638092.6; and the force TM LM or 2PK in the moon s syzygies
is double that quantity. But, descending to the surface of the earth, these
forces are diminished in proportion of the distances from the centre of the
earth, that is, in the proportion of 60| to 1
; and therefore the former force
on the earth s surface is to the force of gravity as 1 to 38604600 ; and by
this force the sea is depressed in such places as are 90 degrees distant from
the sun. But by the other force, which is twice as great, the sea is raised
not only in the places directly under the sun, but in those also which are
directly opposed to it
; and the sum of these forces is to the force of gravity
as 1 to 12868200. And because the same force excites the same motion,
whether it depresses the waters in those places which are 90 degrees distant
from the sun, or raises them in the places which are directly under and di
rectly opposed to the sun, the aforesaid sum will be the total force of the
sun to disturb the sea, and will have the same effect as if the whole was
employed in raising the sea in the places directly under and directly op
posed to the sun, and did not act at all in the places which are 90 degrees
removed from the sun.
And this is the force of the sun to disturb the sea in any given place,
where the sun is at the same time both vertical, and in its mean distance
from the earth. In other positions of the sun, its force to raise the sea is
as the versel sine of double its altitude above the horizon of the place di
rectly, and the of the distance from the earth reciprocally.
COR. Since the centrifugal force of the parts of the earth, arising from
the earth s diurnal motion, which is to the force of gravity as 1 to 289,
raises the waters under the equator to a height exceeding that under the
poles by 85472 Paris feet, as above, in Prop. XIX., the force of the sun,
which we have now shewed to be to the force of gravity as 1 to 12868200,
and therefore is to that centrifugal force as 289 to 12868200, or as 1 to
44527, will be able to raise the waters in the places directly under and di
rectly opposed to the sun to a height exceeding that in the places which arc
90 degrees removed from the sun only by one Paris foot and 113 V inches ;
for this measure is to the measure of 85472 feet as 1 to 44527.
PROPOSITION XXXVII. PROBLEM XVIIL
Tofind the force of the moon to move the sea.
The force of the moon to move the sea is to be deduced from its proper-
29
450 THE MATHEMATICAL PRINCIPLES [BoOH III
tion to the force of the sun, and this proportion is to he collected from the
proportion of the motions of the sea, which are the effects of those forces.
Before the mouth of the river Avon, three miles below Bristol, the height
of the ascent of the water in the vernal and autumnal syzygies of the lu
minaries (by the observations of Samuel Sturmy} amounts to about 45
feet, but in the quadratures to 25 only. The former of those heights ari
ses from the sum of the aforesaid forces, the latter from their difference.
If, therefore, S and L are supposed to represent respectively the forces of
the sun arid moon while they are in the equator, as well as in their mean
distances from the earth, we shall have L + S to L S as 45 to 25, or as
9 to 5.
At Plymouth (by the observations of Samuel Colepress) the tide in its
mean height rises to about 16 feet, and in the spring and autumn tluheight
thereof in the syzygies may exceed that in the quadratures by more
than 7 or 8 feet. Suppose the greatest difference of those heights to be 9
feet, and L -f S will be to L S as 20 to ll|, or as 41 to 23; a pro
portion that agrees well enough with the former. But because of the great
tide at Bristol, we are rather to depend upon the observations of Sturmy ;
and, therefore, till we procure something that is more certain, we shall use
the proportion of 9 to 5.
But because of the reciprocal motions of the waters, the greatest tides do
not happen at the times of the syzygies of the luminaries, but, as we have
said before, are the third in order after the syzygies ;
or (reckoning from
the syzygies) follow next after the third appulse of the moon to the me
ridian of the place after the syzygies ; or rather (as Sturmy observes) are
the third after the day of the new or full moon, or rather nearly after the
twelfth hour from the new or full moon, and therefore fall nearly upon the
forty-third hour after the new or full of the moon. But in this port they
fall out about the seventh hour after the appulse of the moon to the me
ridian of the place ; and therefore follow next after the appulse of the
moon to the meridian, when the moon is distant from the sun, or from op
position with the sun by about IS or 19 degrees in. consequent-la. So the
summer and winter seasons come not to their height in the solstices them
selves, but when the sun is advanced beyuni the solstices by about a tenth
part of its whole course, that is, by about 36 or 37 degrees. In like man
ner, the greatest tide is raised after the appulse of the moon to the meridian
of the place, when the moon has passed by the sun, or the opposition thereof.
by .about a tenth part of the whole motion from one greatest tide to the
next following greatest tide. Suppose that distance about 18^ degrees:
and the sun s force in this distance of the moon from the syzygies and
quadratures will be of less moment to augment and diminish that part o1
the motion of the sea which proceeds from the motion of the moon than in
Ihe syzygies and quadratures themselves in the proportion of the radius tu
BOOK III.] OF NATURAL PHILOSOPHY 451
the co-sine of double this distance, or of an angle of 37 degrees ;
that is- in
proportion of 10000000 to 798)355; and, therefore, in the preceding an
alogy, in place of S we must put 0,79863558.
But farther ; the force of tne moon in the quadratures must be dimin
ished, on account of its declination from the equator ;
for the moon in
those quadratures, or rather in 18^ degrees past the quadratures, declines
from the equator by about 23 13 ; and the force of either luminary to
move the sea is diminished as it declines from the equator nearly in the
duplicate proportion of the co-sine of the declination ; and therefore the
force of the moon in those quadratures is only 0.85703271. ; whence we
have L+0,7986355S to 0,8570327L 0,79863558 as 9 to 5.
Farther yet ; the diameters of the orbit in which the moon should move,
setting aside the consideration of eccentricity, are one to the other as 69
to 70 ; and therefore the moon s distance from the earth in the syzygies
is to its distance in the quadratures, c&teris paribus, as 69 to 70 ; and its
distances, when 18i degrees advanced beyond the syzygies, where the great
est tide was excited, and when 18^ degrees passed by the quadratures,
where the least tide was produced, are to its mean distance as 69,098747
and 69,97345 to 69 1. But the force of the moon to move the sea is in
the reciprocal triplicate proportion of its distance ; and therefore its
forces, in the greatest and least of those distances, are to its force in its
mean distance ;is 0.9830427 and 1,017522 to 1. From whence we have
1,0175221, x 0,79863558. to 0,9830427 X 0,8570327L 0,79863558
as 9 to 5 ; and 8 to L as 1 to 4,4815. Wherefore since the force of the
sun is to the force of gravity as 1 to 12868200, the moon s force will be
to the force of gravity as 1 to 2871400.
COR. 1. Since the waters excited by the sun s force rise to the height of
a foot and ll^V inches, the moon s force will raise the same to the height
of 8 feet and 7/ inches ; and the joint forces of botli will raise the same
to the height of 10^ feet
; and when the moon is in its perigee to the
height of 12 i
feet, and more, especially when the wind sets the same way
as the tide. And a force of that quantity is abundantly sufficient to ex
cite all the motions of the sea, and agrees well with the proportion of
those motions; for in such seas as lie free and open from east to west, asiri
the Pacific sea. and in those tracts of the Atlantic and Ethiopia seas
which lie without the tropics, the waters commonly rise to 6, 9,* 12, cr 15
feet
; but in the Pacific sea, which is of a greater depth, as well as- of a
larger extent, the tides are said to be greater than in the Atlantic andi
Ethiopic seas ;
for to have a full tide raised, an extent of sea from east 1 to
west is required of no less than 90 degrees. In the Ethiopic sea, the watersrise
to a less height within the tropics than in the temperate zones, be
cause of the narrowness of the sea between Africa and the southern parts
of America. In the middle of the open sea the waters cannot rise with*
J52 THE MATHEMATICAL PRINCIPLES [BOOK 111,
out falling together, and at the same time, upon both the eastern and west
ern shores, when, notwithstanding, in our narrow seas, they ought to fall
on those shores by alternate turns ; upon which account there is com
monly but a small flood and ebb in such islands as lie far distant from
the continent. On the contrary, in some ports, where to fill and empty
the bays alternately the waters are with great violence forced in and out
through shallow channels, the flood and ebb must be greater than ordinary ;
as at Plymouth and Chepstow Bridge in England, at the mountains of
St. Michael, and the town of Auranches, in Normandy, and at Combaia
and Pegu in the East Indies. In these places the sea is hurried in and
qjit with such violence, as sometimes to lay the shores under water, some
times to leave them dry for many miles. Nor is this force of the influx
and efflux to be broke till it has raised and depressed the waters to 30, 40,
or 50 feet and above. And a like account is to be given of long and shal
low channels or straits, such as the Mugellrniic straits, and those chan
nels which environ England. The tide in such ports and straits, by the
violence of the influx and efflux, is augmented above measure. But on
such shores as lie towards the deep and open sea with a steep descent,
where the waters may freely rise and fall without that precipitation of
influx and efflux, the proportion of the tides agrees with the forces of the
sun and moon.
COR. 2. Since the moon s force to move the sea is to the force of gravity
as 1 to 2871400, it is evident that this force is far less than to appear
sensibly in statical or hydrostatical experiments, or even in those of pen
dulums. It is in the tides only that this force shews itself by any sensi
ble effect.
COR. 3. Because the force of the moon to move the sea is to the like
force of the sun as 4,4815 to 1, and those forces (by Cor. 14, Prop. LXVI,
Book 1) are as the densities of the bodies of the sun and moon and the
cubes of their apparent diameters conjunctly, the density of the moon will
be to the density of the sun as 4,4815 to 1 directly, and the cube of the
moon s diameter to the cube of the sun s diameter inversely ;
that is (see
ing the mean apparent diameters of the moon and sun are 31 161", and
32 12"), as 4891 to 1000. But the density of the sun was to the den
sity of the earth as 1000 to 4000; and therefore the density of the moon
is to the density of the earth as 4891 to 4000, or as 11 to 9. Therefore
the body of the moon is more dense and more earthly than the earth
itself.
COR. 4. And since the true diameter of the moon (from the observations
of astronomers) is to the true diameter of the earth as 100 to 365, the
mass of matter in the moon will be to the mass of matter in the earth as
1 to 39,788.
Cor. 5. And the accelerative gravity on the surface of the moon will be
30OK HI.] OF NATURAL PHILOSOPHY. 453
about three times less than the accelerative gravity on the surface of thr
earth.
COR. 6. And the distance of the moon s centre from the centre of the
earth will be to the distance of the moon s centre from the common centre
of gravity of the earth and moon as 40,783 to 39,788.
COR. 7. And the mean distance of the centre of the moon from the
centre of the earth will be (in the moon s octants) nearly 60f of the great
est semi-diameters of the earth; for the greatest semi- diameter of the
earth was 1 9658600 Paris feet, and the mean distance of the centres of
the earth and moon, consisting of 60| such semi-diameters, is equal to
1187379440 feet. And this distance (by the preceding Cor.) is to the dis
tance of the moon s centre from the common centre of gravity of the
earth and moon as 40.788 to 39,788 : which latter distance, therefore, is
1158268534 feet. And since the moon, in respect of the fixed stars, per
forms its revolution in 27d
. 7h
. 43f , the versed sine of that angle which
the moon in a minute of time describes is 12752341 to the radius
1000,000000,000000; and as the radius is to this versed sine, so are
1158268534 feet to 147706353 feet. The moon, therefore, falling tow
ards the earth by that force which retains it in its orbit, would in one
minute of time describe 147706353 feet
; and if we augment this force
in the proportion of 17Sf to l?7-, we shall have the total force of
gravity at the orbit of the moon, by Cor. Prop. Ill
; and the moon falling
by this force, in one minute of time would describe 14.8538067 feet. And
at the 60th part of the distance of the moon from the earth s centre, that
is, at the distance of 197896573 feet from the centre of the earth, a body
falling by its weight, would, in one second of time, likewise describe
14,8538067 feet. And, therefore, at the distance of 19615800, which
compose one mean serni -diameter of the earth, a heavy body would de
scribe in falling 15,11175, or 15 feet, 1 inch, and 4^ lines, in the same
time. This will be the descent of bodies in the latitude of 45 degrees.
And by the foregoing table, to be found under Prop. XX, the descent in
the latitude of Paris will be a little greater by an excess of about | parts
of a line. Therefore, by this computation, heavy bodies in the latitude of
Paris falling in vacno will describe 15 Paris feet, 1 inch, 4|f lines, very
nearly, in one second of time. And if the gravity be diminished by tak
ing away a quantity equal to the centrifugal force arising in that latitude
."rom the earth s diurnal motion, heavy bodies falling there will describe
in one second of time 15 feet, 1 inch, and l line. And with this velo
city heavy bodies do really fall in the latitude of Paris, as we have shewn
above in Prop. IV and XIX.
COR. 8. The mean distance of the centres of the earth and moon in the
syzygies of the moon is equal to 60 of the greatest semi-diameters of the
earth, subducting only about one 30th par
1
; of a semi- diameter : and in the
45.4 THE MATHEMATICAL PRINCIPLES [BOOK III,
moon s quadratures the mean distance of the same centres is 60f such semidiameters
of the earth ;
for these two distances are to the mean distance oi
the moon in the octants as 69 and 70 to 69|, by Prop. XXVIII.
COR. 9. The mean distance of the centres of the earth and moon in the
syzygies of the moon is 60 mean semi-diameters of the earth, and a 10th
part of one semi-diameter; and in the moon s quadratures the mean dis
tance of the same centres is 61 mean semi- diameters of the earth, subduct
ing one 30th part of one semi-diameter.
COR. 10. In the moon s syzygies its mean horizontal parallax in the lat
itudes of 0. 30, 38, 45, 52, 60, 90 degrees is 57 20", 57 16", 57 14", 57
12", 57 10", 57 8", 57 4", respectively.
In these computations I do not consider the magnetic attraction of the
earth, whose quantity is very small and unknown : if this quantity should
ever be found out, and the measures of degrees upon the meridian, the
lengths of isochronous pendulums in different parallels, the laws of the mo
tions of the sea, and the moon s parallax, with the apparent diameters of
the sun and moon, should be more exactly determined from phenomena : wo
should then be enabled to bring this calculation to a greater accuracy.
PROPOSITION XXXVIII. PROBLEM XIX.
To find the figure of the moon s body.
If the moon s body were fluid like our sea, the force of the earth to raise
that fluid in the nearest and remotest parts would be to the force of the
moon by which our sea is raised in the places under and opposite to the
moon as the accelerative gravity of the moon towards the earth to the accelerative
gravity of the earth towards the moon, and the diameter of the
moon to the diameter of the earth conjunctly ; that is, as 39,788 to 1, and
100 to 365 conjunctly, or as 1081 to 100. Wherefore, since our sea, by
the force of the moon, is raised to Sf feet, the lunar fluid would be raised
by the force of the earth to 93 feet
; and upon this account the figure of
the moon would be a spheroid, whose greatest diameter produced would
pass through the centre of the earth, and exceed the diameters perpendicu
lar thereto by 186 feet. Such a figure, therefore, the moon affects, and
must have put on from the beginning. Q.E.I.
COR. Hence it is that the same face of the moon always respects the
earth ;
nor can the body of the moon possibly rest in any other position,
but would return always by a libratory motion to this situation ; but those
librations, however, must be exceedingly slow, because of the weakness of
the forces which excite them ; so that the face of the moon, which should
be always obverted to the earth, may, for the reason assigned in Prop. XVI I.
be turned towards the other focus of the moon s orbit, without being im
