ferent planes ; and the force LM, acting in the direction of the line PT
situate in the plane of the orbit PAB, will have the same effect as before
;
neither will it draw the body P from the plane of its orbit. But the other
force NM acting in the direction of a line parallel to ST (and which, there
fore, when the body S is without the line of the nodes is inclined to the
plane of the orbit PAB), besides the perturbation of the motion just now
spoken of as to longitude, introduces another perturbation also as to latitude,
attracting the body P out of the plane of its orbit. And this perturbation,
in any given situation of the bodies P and T to each other, will be as the
generating force MN ; and therefore becomes least when the force MN is
least, that is (as was just now shewn), where the attraction SN is not nrirb
greater nor much less than the attraction SK. Q.E.D.
SK-C. XL] OF NATURAL PHILOSOPHY. 205
COR. 1. Hence it may be easily collected, that if several less bodies P
8, R, &c.; revolve about a very great body T, the motion of the innermost
revolving body P will be least disturbed by the attractions of the others.
when the great body is as well attracted and agitated by the rest (accord
ing to the ratio of the accelerative forces) as the rest are by each other
mutually.
COR. 2. In a system of three bodies, T, P, S, if the accelerative attrac
tions of any two of them towards a third be to each other reciprocally as the
squares of the distances, the body P, by the radius PT, will describe its area
about the body T swifter near the conjunction A and the opposition B than it
will near the quadratures C arid D. For every force with which the body P
is acted on and the body T is not, and which does not act in the direction of
the line PT, does either accelerate or retard the description of the area,
according as it is directed, whether in consequentia or in cwtecedentia.
Such is the force NM. This force in the passage of the body P frcm C
to A is directed in consequentia to its motion, and therefore accelerates
it; then as far as D in atttecedentia, and retards the motion; then in, con
sequentia as far as B ; and lastly in antecedentia as it moves from B to C.
COR. 3. And from the same reasoning it appears that the body P ccBteris
paribuSj moves more swiftly in the conjunction and opposition than in the
quadratures.
COR. 4. The orbit of the body P, cc&teris paribus, is more curve at the
quadratures than at the conjunction and opposition. For the swifter
bodies move, the less they deflect from a rectilinear path. And besides the
force KL, or NM, at the conjunction and opposition, is contrary to the
force with which the body T attracts the body P, and therefore diminishes
that force ; but the body P will deflect the less from a rectilinear path the
less it is impelled towards the body T.
COR. 5. Hence the body P, cceteris paribus, goes farther from the body
T at the quadratures than at the conjunction and opposition. This is said,
E C_ L
B
however, supposing no regard had to the motion of eccentricity. For if
the orbit of the body P be eccentrical, its eccentricity (as will be shewn
presently by Cor. 9) will be greatest when the apsides are in the syzygies;
and thence it may sometimes come to pass that the body P. in its
near approach to the farther apsis, may go farther from the body T at the
syzygies than at the quadratures.
COR. 6. Because the centripetal force of the central body T, by which
206 THE MATHEMATICAL PRINCIPLES [BOOK. 1
the body P is retained in its orbit, is increased at the quadratures by tho
addition caused by the force LM, and diminished at the syzygies by the
subduction caused by the force KL, and, because the force KL is greater
than LM, it is more diminished than increased ; and, moreover, since that
centripetal force (by Cor. 2, Prop. IV) is in a ratio compounded of the sim
ple ratio of the radius TP directly, and the duplicate ratio of the periodi
cal time inversely ;
it is plain that this compounded ratio is diminished by
the action of the force KL ; and therefore that the periodical time, supposing
the radius of the orbit PT to remain the same, will be increased, and that
in the subduplicate of that ratio in which the centripetal force is diminish
ed ; and, therefore, supposing this radius increased or diminished, the peri
odical time will be increased more or diminished less than in the sesquiplicate
ratio of this radius, by Cor. 6, Prop. IV. If that force of the central
body should gradually decay, the body P being less and less attracted would
go farther and farther from the centre T ; and, on the contrary, if it were
increased, it would draw nearer to it. Therefore if the action of the distant
body S, by which that force is diminished, were to increase and decrease
by turns, the radius TP will be also increased and diminshed by turns ;
and the periodical time will be increased and diminished in a ratio com
pounded of the sesquiplicate ratio of the radius, and of the subduplicate oi
that ratio in which the centripetal force of the central body T is dimin
ished or increased, by the increase or decrease of the action of the distant
body S.
COR. 7. It also follows, from what was before laid down, that the axis
of the ellipsis described by the body P, or the line of the apsides, does as
to its angular motion go forwards and backwards by turns, but more for
wards than backwards, and by the excess of its direct motion is in the
whole carried forwards. For the force with which the body P is urged to
the body T at the quadratures, where the force MN vanishes, is compound
ed of the force LM and the centripetal force with which the body T at
tracts the body P. The first force LM, if the distance PT be increased, is
increased in nearly the same proportion with that distance, and the other
force decreases in the duplicate ratio of the distance ; and therefore the
sum of these two forces decreases in a less than the duplicate ratio of the
distance PT ;
and therefore, by Cor. 1, Prop. XLV, will make the line of
the apsides, or, which is the same thing, the upper apsis, to go backward.
But at the conjunction and opposition the force with which the body P is
urged towards the body T is the difference of the force KL, and of the
force with which the body T attracts the body P ; and that difference, be
cause the force KL is very nearly increased in the ratio of the distance
PT, decreases in more -than the duplicate ratio of the distance PT ; and
therefore, by Cor. 1, Prop. XLV, causes the line of the apsides to go for
wards. In the places between the syzygies and the quadratures, the motion
SEC. Xl.J OF NATURAL PHILOSOPHY. 207
of the line of the apsides depends upon both < f these causes conjuncdy, so
that it either goes forwards or backwards in proportion to the excess ol
one of these causes above the other. Therefore since the force KL in the
syzygies is almost twice as great as the force LM in the quadratures, the
excess will be on the side of the force KL, and by consequence the line of
the apsides will be carried forwards. The truth of this arid the foregoing
IE
Corollary will be more easily understood by conceiving the system of the
two bodies T and P to be surrounded on every side by several bodies S,
S, S, dec., disposed about the orbit ESE. For by the actions of these bo
dies the action of the body T will be diminished on every side, and decrease
in more than a duplicate ratio of the distance.
COR. 8. IJut since the progress or regress of the apsides depends upon
the decrease of the centripetal force, that is, upon its being in a greater or
less ratio than the duplicate ratio of the distance TP, in the passage of
the body from the lower apsis to the upper ; and upon a like increase in
its return to the lower apsis again ; and therefore becomes greatest where
the proportion of the force at the upper apsis to the force at the lower ap
sis recedes farthest from the duplicate ratio of the distances inversely ;
it
is plain, that, when the apsides are in the syzygies, they will, by reason of
the subducting force KL or NM LM, go forward more swiftly ; and in
the quadratures by the additional force LM go backward more slowly.
Because the velocity of the progress or slowness of the regress is continued
for a long time ;
this inequality becomes exceedingly great.
COR. 9. If a body is obliged, by a force reciprocally proportional to the
square of its distance from any centre, to revolve in an ellipsis round that
centre ; and afterwards in its descent from the upper apsis to the lower
apsis, that force by a perpetual accession of new force is increased in more
than a duplicate ratio of the diminished distance ;
it is manifest that the
body, being impelled always towards the centre by the perpetual accession
of this new force, will incline more towards that centre than if it were
urged by that force alone which decreases in a duplicate ratio of the di
minished distance, and therefore will describe an orbit interior to that
elliptical orbit, and at the lower apsis approaching nearer to the centre
than before. Therefore the orbit by the accession of this new force will
become more eccentrical. If now, while the body is returning from the
lower to the upper apsis, it should decrease by the same degrees by which
it increases before the body would return to its first distance; and there
THE MATHEMATICAL PRINCIPLES [BOOK I.
fore if the force decreases in a yet greater ratio, the body, being now less
attracted than before, will ascend to a still greater distance, and so the ec
centricity of the orbit will be increased still more. Therefore if the ratio
of the increase and decrease of the centripetal force be augmented each
revolution, the eccentricity will be augmented also ; and, on the contrary,
if that ratio decrease, it will be diminished.
Now, therefore, in the system of the bodies T, P, S, when the apsides of
the orbit FAB are in the quadratures, the ratio of that increase and de
crease is least of all, and becomes greatest when the apsides are in the
syzygies. If the apsides are placed in the quadratures, the ratio near the
apsides is less, and near the syzygies greater, than the duplicate ratio of the
distances : and from that Greater ratio arises a direct motion of the line of
7 o
the apsides, as was just now said. But if we consider the ratio of the
whole increase or decrease in the progress between the apsides, this is less
than the duplicate ratio of the distances. The force in the lower is to the
force in the upper apsis in less than a duplicate ratio of the distance of the
upper apsis from the focus of the ellipsis to the distance of the lower apsis
from the same focus ; and, contrariwise, when the apsides are placed in the
syzygies, the force in the lower apsis is to the force in the upper apsis in a
greater than a duplicate ratio of the distances. For the forces LM in the
quadratures added to the forces of the body T compose forces in a less ra
tio
; and the forces KL in the syzygies subducted from the forces of the
body T, leave the forces in a greater ratio. Therefore the ratio of the
whole increase and decrease in the passage between the apsides is least at
the quadratures and greatest at the syzygies ; and therefore in the passage
of the apsides from the quadratures to the syzygies it is continually aug
mented, and increases the eccentricity of the ellipsis ; and in the passage
from the syzygies to the quadratures it is perpetually decreasing, and di
minishes the eccentricity.
COR. 10. That we may give an account of the errors as to latitude, let
us suppose the plane of the orbit EST to remain immovable; and from
the cause of the errors above explained, it is manifest, that, of the two
forces NM, ML, which are the only and entire cause of them, the force
ML acting always in the plane of the orbit PAB never disturbs the mo
tions as to latitude ; and that the force NM, when the nodes are in the
gyzygies, acting also in the same plane of the orbit, does not at that time
affect those motions. But when the nodes are in the quadratures, it dis
turbs tliem very much, and, attracting the body P perpetually out of the
plane of its orbit, it diminishes the inclination of the plane in the passage
of the body from the quadratures to the syzygies, and again increases the
same in the passage from the syzygies to the quadratures. Hence it
comes to pass that when the body is in the syzygies, the inclination is
then least of all, and returns to the first magnitude nearly, when the body
SEC. XL] OF NATURAL PHILOSOPHY. 209
arrives at the next node. But if the nodes are situate at the octants after
the quadratures, that is, between C and A, D and B, it will appear, from
ii C L
E
wnat was just now shewn, that in the passage of the body P from either
node to the ninetieth degree from thence, the inclination of the plane is
perpetually diminished ; then, in the passage through the next 45 degrees
to the next quadrature, the inclination is increased ; and afterwards, again,
in its passage through another 45 degrees to the next node, it is dimin
ished. Therefore the inclination is more diminished than increased, and
is therefore always less in the subsequent node than in the preceding one.
And, by a like reasoning, the inclination is more increased than diminish
ed when the nodes are in the other octants between A and D, B and C.
The inclination, therefore, is the greatest of all when the nodes are in the
syzygies In their passage from the syzygies to the quadratures the incli
nation is diminished at each appulse of the body to the nodes : and be
comes least of all when the nodes are in the quadratures, and the body in
the syzygies ; then it increases by the same degrees by which it decreased
before ; and, when the nodes come to the next syzygies, returns to its
former magnitude.
COR. 11. Because when the nodes are in the quadratures the body P is
perpetually attracted from the plane of its orbit ; and because this attrac
tion is made towards S in its passage from the node C through the con
junction A to the node D ; and to the contrary part in its passage from the
node D through the opposition B to the node C; it is manifest that, in its
motion from the node C, the body recedes continually from the former
plane CD of its orbit till it comes to the next node; and therefore at that
node, being now at its greatest distance from the first plane CD, it will
pass through the plane of the orbit EST not in D, the other node of that
plane, but in a point that lies nearer to the body S, which therefore be
comes a new place of the node in, antecedentia to its former place. And,
by a like reasoning, the nodes will continue to recede in their passage
from this node to the next. The nodes, therefore, when situate in the
quadratures, recede perpetually ; and at the syzygies, where no perturba
tion can be produced in the motion as to latitude, are quiescent : in the in
termediate places they partake of both conditions, and recede more slowly ;
and, therefore, being always either retrograde or stationary, they will be
carried backwards, or in atitecedentia, each revolution.
COR. 12. All the errors described in these corrollaries arc a little greater
14
210 THE MATHEMATICAL PRINCIPLES BOOK L
at the conjunction of the bodies P, S, than at their opposition ; because
the generating forces NM and ML are greater.
COR. 13. And since the causes and proportions of the errors and varia
tions mentioned in these Corollaries do not depend upon the magnitude of
the body S, it follows that all things before demonstrated will happen, if
the magnitude of the body S be imagined so great as that the system of the
two bodies P and T may revolve about it. And from this increase of the
body S, and the consequent increase of its centripetal force, from which the
errors of the body P arise, it will follow that all these errors, at equal dis
tances, will be greater in this case, than in the other where the body S re
volves about the system of the bodies P and T.
COR. 14. But since the forces NM, ML, when the body S is exceedingly
distant, are very nearly as the force SK and the ratio PT to ST conjunctly
;
that is, if both the distance PT, and the absolute force of the body
8 be given, as ST 3
reciprocally : and since those forces NM, ML are the
causes of all the errors and effects treated of in the foregoing Corollaries;
it is manifest that all those effects, if the system of bodies T and P con
tinue as before, and only the distance ST and the absolute force of the body
S be changed, will be very nearly in a ratio compounded of the direct ratio
of the absolute force of the body S, and the triplicate inverse ratio of the
distance ST. Hence if the system of bodies T and P revolve about a dis
tant body S, those forces NM, ML, and their eifl ts, will be (by Cor. 2 and
6, Prop IV) reciprocally in a duplicate ratio c/f the periodical time. And
thence, also, if the magnitude of the bodv S be proportional to its absolute
force, those forces NM, ML, and their effects, will be directly as the cube
of the apparent diameter of the distant body S viewed from T, and so vice
versa. For these ratios are the same as the compounded ratio above men
tioned.
COR. 15. And because if the orbits ESE and PAB, retaining their fig
ure, proportions, and inclination to each other, should alter their magni
tude ;
arid the forces of the bodies S and T should either remain, or be
changed in any given ratio ; these forces (that is, the force of the body T,
which obliges the body P to deflect from a rectilinear course into the orbit
PAB, and the force of the body S, which causes the body P to deviate from
that orbit) would act always in the same manner, and in the same propor
tion : it follows, that all the effects will be similar and proportional, arid
the times of those effects proportional also
; that is, that all the linear er
rors will be as tne diameters of the orbits, the angular errors the same as
before ; and the times of similar linear errors, or equal angular errors? as
the periodical times of the orbits.
COR. 16. Therefore if the figures of the orbits and their inclination to
each other be given, and the magnitudes, forces, arid distances of the bodies
he any how changed, we may. from the errors and times of those errors in
SEC. XI.] OF NATURAL PHILOSOPHY. 2 \\
one case, collect very nearly the errors and times of the errors in any other
case. But this may be done more expeditiously by the following method.
The forces NM; ML, other things remaining unaltered, are as the radius
TP ; and their periodical effects (by Cor. 2, Lein. X) are as the forces and
the square of the periodical time of the body P conjunctly. These are the
linear errors of the body P ; and hence the angular errors as they appear
from the centre T (that is, the motion of the apsides and of the nodes, and all
the apparent errors as to longitude and latitude) are in each revolution of
the body P as the square of the time of the revolution, very nearly. Let
these ratios be compounded with the ratios in Cor. 14, and in any system
of bodies T, P, S, where P revolves about T very near to it, and T re
volves about S at a great distance, the angular errors of the body P, ob
served from the centre T, will be in each revolution of the body P as the
square of the periodical time of the body P directly, and the square of the
periodical time of the body T inversely. And therefore the mean motion
of the line of the apsides will be in a given ratio to the mean motion of
the nodes ; and both those motions will be as the periodical time of the
body P directly, and the square of the periodical time of the body T in
versely. The increase or diminution of the eccentricity and inclination of
the orbit PAB makes no sensible variation in the motions of the apsides*
and nodes, unless that inc/case or diminution be very great indeed.
COR. 17. Sines the line LM becomes sometimes greater and sometimes
less than the radius PT, let the mean quantity of the force LM be expressed
E C
sa --::-..::::::;
by that radius PT ; and then that mean force will be to the mean force
SK or SN (which may be also expressed by ST) as the length PT to the
length ST. But the mean force SN or ST, by which the body T is re
tained in the orbit it describes about S, is to the force with which the body P
is retained in its orbit about T in a ratio compounded of the ratio of the
radius ST to the radius PT, and the duplicate ratio of the periodical time
of the body P about T to the periodical time of the body T about S. And,
ex cequo, the mean force LM is to the force by which the body P is retain
ed in its orbit about T (or by which the same body P might revolve at the
distance PT in the same periodical time about any immovable point T) in
the same duplicate ratio of the periodical times. The periodical times
therefore being given, together with the distance PT, the mean force LM
is also given ; and that force being given, there is given also the force MN,
very nearly, by the analogy of the lines PT and MN.
212 THE MATHEMATICAL PRINCIPLES [BoOK I
Con. IS. By tlie same laws by which the body P revolves about the
body T, let us suppose many fluid bodies to move round T at equal dis
tances from it
; and to be so numerous, that they may all become contiguous
to each other, so as to form a fluid annulus, or ring, of a round figure, and
concentrical to the body T; and the several parts of this annulus, perform
ing their motions by the same law as the body P, will draw nearer to the
body T, and move swifter in the conjunction and opposition of themselves
and the body S, than in the quadratures. And the nodes of this annulus,
or its intersections with the plane of the orbit of the body S or T, will rest
at the syzygies ; but out of the syzygies they will be carried backward, or
in. antecedentia ; with the greatest swiftness in the quadratures, and more
slowly in other places. The inclination of this annulus also will vary, and
its axis will oscillate each revolution, and when the revolution is completed
will return to its former situation, except only that it will be carried round
a little by the precession of the nodes.
COR. 19. Suppose now the spherical body T, consisting of some matter
not fluid, to be enlarged, and to extend its If on every side as far as that
annulus, and that a channel were cut all round its circumference contain
ing water j and that this sphere revolves uniformly about its own axis in
the same periodical time. This water being accelerated and retarded by
turns (as in the last Corollary), will be swifter at the syzygies, and slower
at the quadratures, than the surface of the globe, and so will ebb and flow in
its channel after the manner of the sea. If the attraction of the body S were
taken away, the water would acquire no motion of flux and reflux by revolv-
.ng round the quiescent centre of the globe. The case is the same of a globe
moving uniformly forwards in a right line, and in the mean time revolving
about its centre (by Cor. 5 of the Laws of Motion), and of a globe uni
formly attracted from its rectilinear course (by Cor. 6, of the same Laws).
But let the body S come to act upon it, and by its unequable attraction the
A\ater will receive this new motion ;
for there will be a stronger attraction
upon that part of the water that is nearest to the body, and a weaker upon
that part which is more remote. And the force LM will attract the water
downwards at the quadratures, and depress it as far as the syzygies ; and the
force KL will attract it upwards in the syzygies, and withhold its descent,
and make it rise as far as the quadratures ; except only in so far as the
motion of flux and reflux may be directed by the channel of the water, and
be a little retarded by friction.
COR. 20. If, now, the annulus becomes hard, and the globe is diminished,
the motion of flux and reflux will cease ; but the oscillating motion of the
inclination and the praecession of the nodes will remain. Let the globe
have the same axis with the annulus, and perform its revolutions in the
same times, and at its surface touch the annulus within, and adhere to it;
then the globe partaking of the motion of the annulus, this whole compares
SEC. XI. OF NATURAL PHILOSOPHY. 213
will oscillate, and the nodes will go backward, for the globe, as \ve shall
shew presently, is perfectly indifferent to the receiving of all impressions.
The greatest angle of the inclination of the annulus single is when the
nodes are in the syzygies. Thence in the progress of the nodes to the
quadratures, it endeavours to diminish its inclination, and by that endea
vour impresses a motion upon the whole globe. The globe retains this
motion impressed, till the annulus by a contrary endeavour destroys that
motion, and impresses a new motion in a contrary direction. And by this
means the greatest motion of the decreasing inclination happens when the
nodes are in the quadratures; and the least angle of inclination in the octants
B
after the quadratures ; and, again, the greatest motion of roclination happens
when the nodes are in the syzygies ; and the greatest angle of reclination in
the octants following. And the case is the same of a globe without this an
nulus, if it be a little higher or a little denser in the equatorial than in the
polar regions : for the excess of that matter in the regions near the equator
supplies the place of the annulus. And though we should suppose the cen
tripetal force of this globe to be any how increased, so that all its parts
were to tend downwards, as the parts of our earth gravitate to the centre,
yet the phenomena of this and the preceding Corollary would scarce be al
tered ; except that the places of the greatest and least height of the water
will be different : for the water is now no longer sustained and kept in its
orbit by its centrifugal force, but by the channel in which it flows. And,
besides, the force LM attracts the water downwards most in the quadra
tures, and the force KL or NM LM attracts it upwards most in the
syzygies. And these forces conjoined cease to attract the water downwards,
and begin to attract it upwards in the octants before the syzygies ; and
cease to attract the water upwards, and begin to attract the water down
