voyage to America, determined that in the island of Cayenne and Granada
the length of the pendulum vibrating in seconds was a small matter less
than 3 feet and 6| lines ; that in the island of St. Christophers it was
3 feet and 6f lines
;
and in the island of St. Domingo 3 feet and 7
lines.
And in the year 1704, P. Feuille, at Puerto Bello in America, found
that the length of the pendulum vibrating in seconds was 3 Paris feet,
and only 5--^ lines, that is, almost 3 lines shorter than at Paris ; but the
observation was faulty. For afterward, going to the island of Martinico.
he found the length of the isochronal pendulum there 3 Paris feet and
5 \ | lines.
Now the latitude of Paraiba is 6 38 south ; that of Puerto Bello 9
33 north ; and the latitudes of the islands Cayenne, Goree, Gaudaloupe}
Martinico, Granada, St. Christophers, and St. Domingo, are respectively
4C 55 , 14 40", 15 00 , 14 44 , 12 06 , 17 19 , and 19 48 , north. An*J
412 THE MATHEMATICAL PRINCIPLES [BOOK III
the excesses of the length of the pendulum at Paris above the lengths of
the isochronal pendulums observed in those latitudes are a little greater
than by the table of the lengths of the pendulum before computed. And
therefore the earth is a little higher under the equator than by the prece
ding calculus, and a little denser at the centre than in mines near the sur
face, unless, perhaps, the heats of the torrid zone have a little extended the
length of the pendulums.
For M. Picart has observed, that a rod of iron, which in frosty weather
in the winter season was one foot long, when heated by lire, was lengthened
into one foot and -]-
line. Afterward M. de la Hire found that a rod of
iron, which in the like winter season was 6 feet long, when exposed to the
heat of the summer sun, was extended into 6 feet and f line. In the former
case the heat was greater than in the latter
; but in the latter it was greater
than the heat of the external parts of a human body ; for metals exposed
to the summer sun acquire a very considerable degree of heat. But the rod
of a pendulum clock is never exposed to the heat of the summer sun, nor
ever acquires a heat equal to that of the external parts of a human body ;
and, therefore, though the 3 feet rod of a pendulum clock will indeed be a
little longer in the summer than in the winter season, yet the difference will
scarcely amount to \ line. Therefore the total difference of the lengths of
isochronal pendulums in different climates cannot be ascribed to the differ
ence of heat ; nor indeed to the mistakes of the French astronomers. For
although there is not a perfect agreement betwixt their observations, yet
the errors are so small that they may be neglected ; and in this they all
agree, that isochronal pendulums are shorter under the equator than
at the Royal Observatory of Paris, by a difference not less than 1{ line,
nor greater than 2| lines. By the observations of M. Richer, in the island
of Cayenne, the difference was 1| line. That difference being corrected by
those of M. des Hayes, becomes \\ line or l line. By the less accurate
observations of others, the same was made about two lines. And this dis
agreement might arise partly from the errors of the observations, partly
from the dissimilitude of the internal parts of the earth, and the height of
mountains ; partly from the different heats of the air.
I take an iron rod of 3 feet long to be shorter by a sixth part of one line
in winter time with us here in England than in the summer. Because of
the great heats under the equator, subduct this quantity from the difference
of one line and a quarter observed by M. Richer, and there will remain one
line TV, which agrees very well with l T -oo ^ne collected, by the theory a
little before. M. Richer repeated his observations, made in the island of
Cayenne, every week for ten months together, and compared the lengths of
the pendulum which he had there noted in the iron rods with the lengths
thereof which he observed in Prance. This diligence and care seems to
have been wanting to the other observers. If this gentleman s observations
BOOK I1I.J OF NATURAL PHILOSOPHY. 413
are to be depended on, the earth is higher under the equator than at the
poles, and that by an excess of about 17 miles; as appeared above by the
theory.
PROPOSITION XXI. THEOREM XVII.
That the equinoctial points go backward, and that the axis of the earth,
by a nutation in, every annual revolution, twice vibrates towards the
ecliptic, and as often returns to its former position,.
The proposition appears from Cor. 20, Prop. LXVI, Book I
; but
that motion of nutation must be very small, and, indeed, scarcely per
ceptible.
PROPOSITION XXII. THEOREM XVIII.
That all the motions of the ?noon, and all the inequalities of those motions,
follow from the principles which we have laid down.
That the greater planets, while they are carried about the sun, may in
the mean time carry other lesser planets, revolving about them ; and that
those lesser planets must move in ellipses which have their foci in the cen
tres of the greater, appears from Prop. LXV, Book I. But then their mo
tions will be several ways disturbed by the action of the sun, and they will
suffer such inequalities as are observed in our moon. Thus our moon (by
Cor. 2, 3, 4, and 5, Prop. LXVI, Book I) moves faster, and, by a radius
drawn to the earth, describes an area greater for the time, and has its orbit
less curved, and therefore approaches nearer to the earth in the syzygies
than in the quadratures, excepting in so far as these effects are hindered by
the motion of eccentricity ;
for (by Cor. 9, Prop. LXVI, Book I) the eccen
tricity is greatest when the apogeon of the moon is in the syzygies, and
least when the same is in the quadratures ; and upon this account the perigeon
moon is swifter, and nearer to us, but the apogeon moon slower,
arid farther from us, in the syzygies than in the quadratures. Moreover,
the apogee goes forward, and the nodes backward ; and this is done not with
a regular but an unequal motion. For (by Cor. 7 and 8, Prop. LXVI,
Book I) the apogee goes more swiftly forward in its syzygies, more slowly
backward in its quadratures; and, by the excess of its progress above its
regress, advances yearly in consequentia. But, contrariwise, the nodes (by
Cor. 11, Prop. LXVI, Book I) are quiescent in their syzygies, and go fastest
back in their quadratures. Farther, the greatest latitude of the moon (by
Cor. 10, Prop. LXVI, Book I) is greater in the quadratures of the moon
than in its syzygies. And (by Cor. 6, Prop. LXVI, Book I) the mean mo
tion of the moon is slower in the perihelion of the earth than in its aphelion.
And these are the principal inequalities (of the moon) taken notice of by
astronomers.
414 THE MATHEMATICAL PRINCIPLES [BOOK III
But there are yet other inequalities not observed by former astronomers,
by which the motions of the moon are so disturbed, that to this day we
have not been able to bring them under any certain rule. For the veloc
ities or horary motions of the apogee and nodes of the moon, and their
equations, as well as the difference betwixt the greatest eccentricity in the
syzygics, and the least eccentricity in the quadratures, and that inequality
which we call the variation, are (by Cor. 14, Prop. LXVI, Book I) in the
course of the year augmented and diminished in the triplicate proportion
of the sun s apparent diameter. And besides (by Cor. 1 and 2, Lem. 10,
and Cor. 16, Prop. LXVI, Book I) the variation is augmented and
diminished nearly in the duplicate proportion of the time between
the quadratures. But in astronomical calculations, this inequality
is commonly thrown into and confounded with the equation of the moon s
centre.
PROPOSITION XXI1L PROBLEM V.
To derive the unequal motions of the satellites of Jupiter and Saturn
from the motions of our moon.
From the motions of our moon we deduce the corresponding motions of
the moons or satellites of Jupiter in this manner, by Cor. 16, Prop. LXVI,
Book I. 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 compound
ed of the duplicate proportion of the periodic times of the earth about the
sun to the periodic times of Jupiter about the sun, and the simple propor
tion of the periodic time of the satellite about Jupiter to the periodic time
of our moon about the earth ; and, therefore, those nodes, in the space of
a hundred years, are carried 8 24 backward, or in antecedentia. 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 the
former, by the same Corollary, and are thence given. And the motion of
the apsis of every satellite in consequential is to the motion of its nodes in
antecedentia as the motion of the apogee of our moon to the motion of its
nodes (by the same Corollary), and is thence given. But the motions of
the apsides thus found must be diminished in the proportion of 5 to 9, or
of about 1 to 2, on account of a cause which I cannot here descend to ex
plain. The greatest equations of the nodes, and of the apsis of every satel
lite, are to the greatest equations of the nodes, and apogee of our moon re
spectively, as the motions of the nodes and apsides of the satellites, in the
time of one revolution of the former equations, to the motions of the nodes
and apogee of our moon, in the time of one revolution of the latter equa
tions. The variation of a satellite seen from Jupiter is to the variation of
our moon in tne same proportion as the whole motions of their node?
BOOK IIIJ OF NATURAL PHILOSOPHY. 415
respectively during the times in which the satellite and our moon (after
parting from) are revolved (again) to the sun, by the same Corollary ; and
therefore in the outmost satellite the variation does not exceed 5" 12 ".
PROPOSITION XXIV. THEOREM XIX.
That the flax and reflux of the sea arise from the actions oj the sun
and moon.
By Cor. 19 and 20, Prop. LXVI, Book I, it appears that the waters of
the sea ought twice to rise and twice to fall every day. as well lunar as solar ;
and that the greatest height of the waters in the open and deep seas ought
to follow the appulse of the luminaries to the meridian of the place by a
less interval than 6 hours ;
as happens in all that eastern tract of the Atlantic
and jEthinpic seas between France and the Cape of Good Hope ; and on
the coasts of Chili and Pern, in the Smith Sea ; in all which shores the
ilo >d falls out about the second, third, or fourth hour, unless where the
motion propagated from the deep ocean is by the shallowness of the chaiir
nels, through which it passes to some particular places, retarded to the
fifth, sixth, or seventh hour, and even later. 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
24th parts uf that time which the moon, by its apparent diurnal motion,
employs to come about again to the meridian of the place which it left the
day before. The force of the sun or moon in raising the sea is greatest in
the appulse of the luminary to the meridian of the place; but the force
impressed upon the sea at that time continues a little while after the im
pression, and is afterwards increased by a new though less force still act
ing upon it. This makes the sea rise higher and higher, till this new force
becoming too weak to raise it any more, the sea rises to its greatest height.
And this will come to pass, perhaps, in one or two hours, but more fre
quently near the shores in about three hours, or even more, where the sea
is shallow.
The two luminaries excite two motions, wrhich will not appear distinctly,
but between them will arise one mixed motion compounded out of both.
In the conjunction or opposition of the luminaries their forces will be con
joined, and bring on the greatest flood and ebb. In the quadratures the
sun will raise the waters which the moon depresses, and depress the waters
which the moon raises, 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 waters
will happen about the third lunar hour. Out of the syzygies and quadra
tures, the greatest tide, which by the single force of the moon oujjht 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 interme
416 THE MATHEMATICAL PRINCIPLES [BOOK in
diate time that aproaches nearer to the third hour of the moon than tc
that of the sun. And, therefore, while the moon is passing from the syzy
gies to the quadratures, during which time the 3d hour of the sun precedes
the 3d hour of the moon, the greatest height of the waters will also precede
the 3d hour of the moon, and that, by the greatest interval, a little after
the octants of the moon; and, by like intervals, the greatest tide will fol
low the 3d lunar hour, while the moon is passing from the quadratures to
the syzygies. Thus it happens in the open sea : for in the mouths of
rivers the ogreater tides come liter to their heiiOrht.
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 diameter. 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
syzygies something greater, and those in the quadratures something less
than in the summer season ; and every month the moon, while in the peri
gee, raises greater tides than at the distance of 15 days before or after,
when it is in its apogee. Whence it comes to pass that two highest
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 force of the moon, then situated in the
equator, most exceeds the force 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 syzy
gies is always succeeded by the least tide in the quadratures, as we find
by experience. But, because the sun is less distant from the earth in
winter than in summer, it comes to pass that the greatest and least tides
more frequently appear before than after the vernal equinox, and more
frequently after than before the autumnal.
Moreover, the effects of the lumi
naries depend upon the latitudes of
places. Let AjoEP represent the
earth covered with deep waters ; C
its centre; P, p its poles; AE the
equator ; F any place without the
equator ; F/ the parallel of the place ;
/F~ M ^ Drl the correspondent parallel on the
K 1ST
BOOK III.] OF NATURAL PHILOSOPHY. 417
other side of the equator; L the place of the moon three Lours 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, Ck, its least heights: and if with
the axes H//, K/.*, an ellipsis is described, and by the revolution of that
ellipsis about its longer axis H/i a spheroid HPKhpk is formed, this sphe
roid will nearly represent the figure of the sea; and CF, C/, CD, Cd,
will represent the heights of the sea in the places F/, Dd. But far
ther ;
in the said revolution of the ellipsis any point N describes the circle
NM cutting the parallels F/, Dd, in any places RT, 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 great
est ebb in Q,, at the third hour after the setting of the moon ; and then
the greatest flood in/, at the third hour after the appulse of the moon to
the meridian under the horizon ; and, lastly, the greatest ebb in Q,, at the
third hour after the rising of the moon ; and the latter flood in / will be
less than the preceding flood in F. For the whole sea is divided into two
hemispherical floods, one in the hemisphere KH/J on the north side, the
other in the opposite hemisphere Khk, which we may therefore call the
northern and the southern floods. These floods, being always opposite the one
to the other, come by turns to the meridians of all places, after an interval
of 12 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 which the luminaries rise and set. But the greatest 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 hori
zon ; and when the moon changes its declination to the other side of the
equator, 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
Hrst of Aries. So it is found by experience that the morning tides in
winter exceed those of the evening, and the evening tides in summer ex
ceed those of the morning ; at Plymouth by the height of one foot, but at
Bristol by the height of 15 inches, according to the observations of Colepress
and Sturmy.
But the motions which we have been describing suffer some alteration
from that force of reciprocation, which the waters, being once moved, retain
a little while by their vis insita. Whence it comes to pass that the tides
may continue for some time, though the actions of the luminaries should
27
418 THE MATHEMATICAL PRINCIPLES [BOOK III
oease. This power of retaining the impressed motion lessens the difference
yf the alternate tides, and makes those tides which immediately succeed
after the syzygies greater, and those which follow next after the quadra
tures less. And hence it is that the alternate tides at Plymouth and
Bristol do not differ much more one from the other than by the height of
a foot or 15 inches, and that the greatest tides of 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 shallow channels, so that the greatest
tides of all, in some straits and mouths of rivers, are the fourth or even the
fifth after the syzygies.
Farther, it may happen 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 mo
tions of different kinds. Let us suppose two equal tides flowing towards
the same port from different places, the one preceding the other by 6 hours ;
and suppose the first tide to happen at the third hour of the appulse of the
moon to the meridian of the port. If the moon at the time of the appulse
to the meridian was in the equator, every 6 hours alternately there would
arise equal floods, which, meeting writh as many equal ebbs, would so bal
ance one the other, that for that day, the water would stagnate 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 thence two
greater and two lesser tides wrould be alternately propagated 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 the two lesser floods the
waters would rise to their least height. Thus in the space of 24 hours the
waters would come, not twice, as commonly, but once only to their great
est, and once only to their least height ; and their greatest height, if the
moon declined towards the elevated pole, would happen at the 6th or 30th
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. An ex
ample of all which Dr. Halley has given us, from the observations of sea
men in the port of Bntshnm, in the kingdom of Tunqvin, 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 flow 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 increases with the declination of
the moon till the ?th or 8th day ; then for the 7 or 8 days following it
BOOK III.] OF NATURAL PHILOSOPHY. 419
decreases at the same rate as it had increased before, and ceases when the
moon changes its declination, crossing over the equator to the south. Af
ter 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 passing the equator, changes its declination. There are
two inlets to this port and the neighboring channels, one from the seas of
China, between the continent and the island of Lenconia ; the other from
the Indian sea, between the continent and the island of Borneo. But
whether there be really two tides propagated through the said channels, one
from the Indian sea in the space of 12 hours, and one from the sea of
Cliina in the space of 6 hours, which therefore happening at the 3d and
9th lunar hours, by being compounded together, produce those motions : or
whether there be any other circumstances in the state of those seas. I leave
to be determined by observations on the neighbouring shores.
Thus I have explained the causes of the motions of the moon and of the
sea. Now it is fit to subjoin something concerning the quantity of those
motions.
PROPOSITION XXV. PROBLEM VI.
To find the forces with which the sun disturbs the motions of the moon.
Let S represent the sun, T the
earth, P the moon, CADB the
moon s orbit. In SP take SK
equal to ST; and let SL be to
SK in the duplicate proportion
of SK to SP: draw LM parallel
to PT ; and if ST or SK is supposed
to represent the accelerated force of gravity of the earth towards the
sun, SL will represent the accelerative force of gravity of the moon towards
the sun. But that force is compounded of the parts SM and LM, of which
the force LM, and that part of SM which is represented by TM, disturb
the motion of the moon, as we have shewn in Prop. LXVI, Book I, and
its Corollaries. Forasmuch as the earth and moon are revolved about
their common centre of gravity, the motion of the earth about that centre
will be also disturbed by the like forces; but we may consider the sums
both of the forces and of the motions as in the moon, and represent the sum
of the forces by the lines TM and ML, which are analogous to them both.
The force ML (in its mean quantity) is to the centripetal force by which
the moon may be retained in its orbit revolving about the earth at rest, at
the distance P J
, in the duplicate proportion of the periodic time of the
moon about the earth to the periodic time of the earth about the sun (by
Cor. 17, Prop. LXVI, Book I) ; that is, in the duplicate proportion of 27 d.
7\ 43 to 365 1
. 6". 9 ; or as 1000 to 178725 ; or as 1 to 178f J. But in the
J23 THE MATHEMATICAL PRINCIPLES [BOOK 111
4ih Prop, of this Book we found, that, if both earth and moon were revolved
aoout their common centre of gravity, the mean distance of the one from
the other would be nearly 60^ mean semi-diameters of the earth : and the
force by which the moon may be kept revolving in its orbit about the earth
in rest at the distance PT of 60^ semi-diameters of the earth, is to the
force by which it may be revolved in the same time, at the distance of 60
semi-diameters, as 60| to 60 : and this force is to the force of gravity with
u,;
3 very nearly as I to 60 X 60. Therefore the mean force ML is to the
force of gravity on the surface of our earth as 1 X 60-} to 60 X 60 X 60
X l~8f, or as 1 to 638092,6 : whence by the proportion of the lines TM,
ML, the force TM is also given; and these are the forces with which the
sun disturbs the motions of the moon. Q.E.I.
PROPOSITION XXVI. PROBLEM VII.
To find the horary increment of the area which the moon, by a radius
