# The Mathematical Principles of Natural Philosophy (1729)/Book 1/Section 9

*Of the motion of bodies in moveable orbits; and of the motion of the apsides.*

Proposition XLIII. Problem XXX.

*It is required to make a body move, in*
a trajectory that revolves about the centre
of force, in the same manner
as another body in the same trajectory
at rest.

In the orbit *VPK* (Pl. 18. *Fig.* 1.) given by
position, let the body P revolve, proceeding from
*V* towards *K*. From the centre *C *let there be
continually drawn *Cp*, equal to *CP*, making the
angle *VCp* proportional to the angle *VCP*; and
the area which the line *Cp* describes, will be to
the area *VCP* which the line *CP* describes at the
same time, as the velocity of the describing line *Cp*, to the velocity of the describing line *CP*; that
is, as the angle *VCp* to the angle *VCP*, therefore
in a given ratio, and therefore proportional
to the time. Since then the area described by
the line *Cp* in an immovable plane is proportional
to the time, it is manifest that a body, being
acted upon by a just quantity of centripetal
force, may revolve with the point *p* in the
curve line which the same point *p*, by the method
just now explained, may be made to describe
in an immovable plane. Make the angle
*VCu* equal to the angle *PCp*, and the line *Cu*
equal to *CV* and the figure *uCp* equal to the
figure *VCP*, and the body being always in the
point *p*, will move in the perimeter of the revolving
figure *uCp*, and will describe its (revolving)
arc *up* in the same time that the other
body P describes the similar and equal arc *VP*
in the quiescent figure *VPK*. Find then by cor.
5. prop. 6. the centripetal force by which a body
may be made to revolve in the curve line
which the point *p* describes in an immovable
plane, and the problem will be solved. *Q. E. F.*

Proposition XLIV. Theroem XIV.

*The difference of the forces, by which two bodies may be made to move equally, one in a quiescent, the other in the same orbit revolving, is in a triplicate ratio of their common altitudes*
inversely.

Let the parts of the quiescent orbit *VP*, *PK*,
(Pl. 18. *Fig.* 2.) be similar and equal to the parts of the revolving orbit *up*, *pk*; and let the
distance of the points *P* and *K* be supposed of the
utmost smallness. Let fall a perpendicular *kr* from
the point *k* to the right line *pC*, and produce it
to *m*, so that *mr* may be to *kr* as the angle *VCp*
to the angle *VCP*. Because the altitudes of the
bodies, *PC* and *pC*, *KC* and *kC*, are always equal,
it is manifest: that the increments or decrements of
the lines *PC* and *pC* are always equal; and therefore
if each of the several motions of the bodies
in the places *P* and *p* be resolved into two, (by
cor. 2. of the laws of motion) one of which is
directed towards the center, or according to the
lines *PC*, *pC*, and the other, transverse to the former,
hath a direction perpendicular to the lines
*PC* and *pC*; the motions towards the centre will
be equal, and the transverse motion of the body
*p* will be to the transverse motion of the body *P*,
as the angular motion of the line *pC* to the angular
motion of the line *PC*; that is, as the angle
*VCp* to the angle *VCP*. Therefore at the same
time that the body *P*, by both its motions, comes
to the point *K*, the body *p*, having an equal motion
towards the centre, will be equally moved
from *p* towards *C*, and therefore that time being
expired, it will be found somewhere in the line
*mkr*, which, passing through the point *k*, is perpendicular
to the line *pC*; and by its transverse
motion, will acquire a distance from the line *pC*,
that will be to the distance which the other body
*P* acquires from the line *PC*, as the transverse
motion of the body *p*, to the transverse motion
of the other body *P*. Therefore since *kr* is equal
to the distance which the body *P* acquires
from the line *PC*, and *mr* is to *kr* as the angle *VCp* to the angle *VCP*, that is as the tranvers
motion of the body *p*, to the transverse motion
of the body *P*: it is manifest that the body *p*,
at the expiration of that time, will be found in the
place *m*. These things will be so, if the bodies *p*
and *P* are equally moved in the directions of the
lines *pC* and *PC*, and are therefore urged with
equal forces in those directions. But if we take
an angle *pCn* that is to the angle *pCk; as the angle*
*VCp* to the angle *VCP*, and *nC* be equal
to *kC* in that case the body *p* at the expiration
of the time will really be in *n*; and is therefore
urged with a greater force than the body *P*, if
the angle *nCp* is greater than the angle *kCp*, that
is, if the orbit *upk* move either *in consequantia, or*
in *antecedentia* with a celerity greater than the
double of that with which the line *CP* moves in
consequentia; and with a less force if the orbit moves
flower in *antecedentia*. And the difference of the
forces will be as the interval *mn* of the places
through which the body would be carried by the
action of that difference in that given space of time.
About the centre *C* with the interval *Cn* or *Ck*
suppose a circle described cutting the lines *mr*, *mn*
produced in *s* and *r*, and the rectangle *mn* x *mt*
will be equal to the rectangle *mk* x *ms*, and therefore
*mn* will be equal to . But since the
triangles *pCk*, *pCn*, in a given time, are of a
given magnitude, *kr* and *mr*, and their difference
*mk*, and their sum *mr*, are reciprocally as the altitude
*pC*, and therefore the rectangle '*mk* x *rm* is
reciprocally as the square of the altitude *pC*. But
moreover *mr* is directly as *ms*, that is, as the altitude *pC*. These are the first ratio's of the nascent
lines; and hence , that is, the nascent lineola
*mn*, and the difference of the forces proportional
thereto, are reciprocally as the cube of the
altitude *pC*. *Q. E. D.*

Cor. 1. Hence the difference of the forces in
the places *P* and *p*, or *K* and *k*, is to the force
with which a body may revolve with a circular
motion from *R* to *K*, in the same time that the
body *P* in an immovable orb describes the arc
*PK*, as the nascent line *mn* to the versed sine of
the nascent arc *RK*, that is as to ,
or as *mk* x *ms* to the square of *rk*; that is,
if we take given quantities *F* and *G* in the same
ratio to one another as the angle *VCP* bears to
the angle *VCp*, as *GG* - *FF* to *FP*. And there.
fore if from the centre *C* with any distance *CP*
or *Cp*, there be described a circular sector equal
to the whole area *VPC*, which the body revolving
in an immovable orbit, has by a radius
drawn to the centre described in any certain time;
the difference of the forces, with which the body
*P* revolves in an immovable orbit and the body
*p* in a moveable orbit, will be to the centripetal
force, with which another body by a radius drawn to
the centre can uniformly describe that sector in
the same time as the area *VPC* is described, as
*CG* - *FF* to *FP*. For that sector and the area
*pCk* are to one another as the times in which they
are described.

Cor. 2. If the orbit *VPK* be an ellipsis having
its focus *C*, and its highest apsis *V*, and we suppose
the ellipsis *upk* similar and equal to it, so that *pC*
may be always equal to *PC*, and the angle *VCp* be
to the angle *VCP' in the given ratio of *G* to *F*;*
and for the altitude *PC* or *pC* we put *A*, and a *R*
for the latus rectum of the ellipsis; the force with
which a body may be made to revolve in a moveable
ellipsis will be as and
vice versa. Let the force with which a body may
revolve in an immovable ellipsis, be expressed by
the quantity , and the force in *V* will be
. But the force with which a body may
revolve in a circle at the distance *CV* with the
same velocity as a body revolving in an ellipsis has
in *V*, is to the force with which a body revolving
in an ellipsis is acted upon in the apsis *V*,
as half the latus rectum of the ellipsis, to the
semi-diameter *CV* of the circle, and therefore is as
; and the force which is to this as CG - FF to FF, is as (by
cor. 1. of this prop.) is the difference of the
forces in *V* with which the body P revolves in
the immovable ellipsis *VPK*, and the body *p* in
the moveable ellipsis *upk*, Therefore since by this
prop. that difference at any other altitude A is to
it self at the altitude *CV* as to , the same difference in every altitude *A* will be as
.
Therefore to the force ,
by which the body may revolve in an immovable
ellipsis *VPK*, add the excess
and the sum will be the whole force by which a body may revolve
in the same time in the moveable ellipsis *upk*.

Cor. 3. In the same manner it will be found
that if the immovable orbit *VPK* be an ellipsis
having its centre in the centre of the forces *C*;
and there be supposed a moveable ellipsis *upkk* similar,
equal, and concentrical to it; and 2 R be the
principal latus reclum of that ellipsis, and 2 T the
latus transversum or greater axis; and the angle *VCp*
be continually to the angle *VCP* as *G* to *F*; the
forces with which bodies may revolve in the immovable
and moveable ellipsis in equal times, will
be as and resperctively.

Cor. 4. And universally, if the greatest altitude
*CV* of the body be called T, and the radius of the
curvature which the orbit *VPK* has in *V*, that is,
the radius of a circle equally curve, be called *R*,
and the centripetal force with which a body may
revolve in any immovable trajectory *VPK* at the
place *K* be called , and in other places *P* be
indefinitely stiled X; and the altitude *CP* be called A, and G be taken to F in the given ratio of the
angle *VCp* to the angle *VCP*: the centripetal
force with which the same body will perform the
same motions in the same time in the same trajectory
*mpk* revolving with a circular motion, will
be as the forces .

Cor. 5.. Therefore the motion of a body in an immovable orbit being given, its angular motion round the centre of the forces may be increased or diminished in a given ratio, and thence new immovable orbits may be found in which bodies may revolve with new centripetal forces.

Cor. 6. Therefore if there be erected (Pl. 18. *Fig.*
3.) the line *VP* of an indeterminate length, perpendicular
to the line *CV* given by position, and
*CP* be drawn, and *Cp* equal to it, making the angle
*VCp* having a given ratio to the angle *VCP*;
the force with which a body may revolve in the
curve line *Vpk*, which the point *p* is continually
describing, will be reciprocally as the cube of the
altitude *Cp*. For the body *P*, by its vis inertiæ
alone, no other force impelling it, will proceed
uniformly in the right line *VP*. Add then a force
tending to the centre *C* reciprocally as the cube of
the altitude *CP* or *Cp*, and (by what was just demonstrated)
the body will deflect from the rectilinear
motion into the curve line *Vpk*, But this curve
*Vpk* is the same with the curve *VPQ* found in cor.
3. prop. 41. in which, I said, bodies attracted with
such forces would ascend obliquely.

Proposition XLV. Problem XXXI.

*To find the motion of the apsides in orbits approaching very near to circles.*

This problem is solved arithmetically by reducing
the orbit, which a body revolving in a moveable
ellipsis (as in cor. 2. and 3 of the above prop.)
describes in an immovable plane, to the figure of
the orbit whose apsides are required; and then
seeking the apsides of the orbit which that body
describes in an immovable plane. But orbits acquire
the same figure, if the centripetal forces with
which they are described, compared between themselves,
are made proportional at equal altitudes. Let
the point *V* be the highest apsis, and write T for
the greatest altitude *CV*, A for an other altitude
*CP* or *Cp*, and X for the difference of the altitudes
*CV* - *CP*, and the force with which a body moves
in an ellipsis revolving about its focus *C* (as in cor. 2.)
and which in cor. 2. was as ,
that is as , by substituting T - X for A will become .
In like manner any other centripetal force is to be
reduced to a fraction whose denominator is A
and the numerators are to be made analogous by collating together the homologous terms. This
will be made plainer by examples.

Exam 1.. Let us suppose the centripetal force
to be uniform, and therefore as , or, writing T - X
for A inthe numerator, as
. Then collating
together the correspondent terms of the numerators,
that is, those that confist of given quantities.
with those of given quantities, and those of
quantities not given, with those of quantities not
given, it will become RGG - EFF + TFF
to as - FFX to
or as - FF to - 3TT + 3TX - XX. Now
since the orbit is supposed extreamly near to a circle,
let it coincide with a circle, and because in
that case R and T become equal, and X is infinitely
diminished, the last ratio's will be, as RGG
to so - FF to 3TT, or as GG to TT
so FF to 3TT, and again as GG to FF so
TT to 3TT, that is, as 1 to 3; and therefore
G is to F, that is, the angle *VCp* to the angle
*VCP* as 1 to, . Therefore since the body, in
an immoveable ellipsis, in descending from the upper
to the lower apsis, describes an angle, if I may
so speak, of 180 deg. the other body in a moveable
ellipsis, and therefore in the immovable orbit
we are treating of, will, in its descent from the
upper to the lower apsis, describe an angle *VCp* of
deg. And this comes to pas by reason of the likeness of this orbit which a body acted upon
by an uniform centripetal force describes, and
of that orbit which a body performing its circuits
in a revolving ellipsis will describe in a quiescent
plane. By this collation of the terms, these orbits
are made similar, not universally indeed, but then
only when they approach very near to a circular
fugure. A body therefore revolving with an uniform
centripetal force in an orbit nearly circular,
will always describe an angle of deg. or 103 deg.
55 m. 23 sec. at the centre; moving from the
upper apsis to the lower apsis when it has once
described that angle, and thence returning to the
upper apsis when it has described that angle again;
and so on in infinitum.

Exam. 2. Suppose the centripetal force to be
as any power of the altitude A, as for example
or ; where *n* - 3 and *n* signify any
indices of powers whatever, whether integers or
fractions, rational or surd, affirmative or negative.
That numerator or being reduced to
an indeterminate series by my method of converging
series, will become &c. And conferring these
terms with the terms of the other numerator
RGG - RFF + TFF - FXX, it becomes as
RGG - RFF + TFF to so - FF to
&c. And taking
the last ratio's where the orbits approach to circles, it becomes as RGG to so -FF to
, or as GG to so FF to
, and again GG to FF so to
, that is, as 1 to *n*; and therefore G is
to F, that is the angle *VCp* to the angle *VCP* as
1 to . Therefore since the angle *VCP*, described
in the descent of the body from the upper
apsis to the lower apsis in an ellipsis, is of
180 deg. the angle *VCp*, described in the descent
of the body from the upper apsis to the lower
apsis in an orbit nearly circular which a body describes
with a centripetal force proportional to the
power , will be equal to an angle of
deg. and this angle being repeated the body
will return from the lower to the upper apsis, and
so on in infinitum. As if the centripetal force be
as the distance of the body from the centre, that
is, as A, or , *n* will be equal to 4 and,
equal to 2; and therefore the angle between the
upper and the lower apsis will be equal to
deg. or 90 deg. Therefore the body having performed
a fourth part of one revolution will arrive
at the lower apsis, and having performed another
fourth part, will arrive at the upper apsis, and so
on by turns in infinitum. This appears also from
prop. 10. For a body acted on by this centripetal
force will revolve in an immovable ellipsis, whose
centre is the centre of force. If the centripetal
force is reciprocally as the distance, that is,
drectly as or as , *n* will be equal to 2, and therefore the angle between the upper and lower
apsis will be of deg. or 127 deg. 16 min. 45
sec. and therefore a body revolving with such a
force, will, by a perpetual repetition of this angle,
move alternately from the upper to the lower, and
from the lower to the upper apsis for ever. So
also if the centripetal force be reciprocally as the
biquadrate root of the eleventh power of the altitude,
that is reciprocally as and therefore
directly as or as , as will be equal to
and deg. will be equal to 360 deg. and therefore
the body parting from the upper apsis, and
from thence perpetually descending will arrive at
the lower apsis when it has compleated one entire
revolution; and thence ascending perpetually,
when it has compleated another entire revolution
it will arrive again as the upper apsis; and so alternately
for ever.

Exam. 3.. Taking *m* and *n* for any indices of
the powers of the altitude, and *b* and *c* for any
given numbers, suppose the centripetal force to be
as
or (by the method of converging series above-mentioned) as
&c.
and comparing the terms of the numerators, there will arise RGC - RFF + TFF to
as -FF to &c.
And taking the last ratio's that arise when the orbits
come to a circular form, there will come
forth GG to as FF to
and again GG to FF
as to
.
This proportion. by expressing the greatest altitude
*CV* or T arithmetically by unity, becomes, GG
to FF as *b* + *c* to *mb* + *nc*, and therefore as 1
to . Whence G becomes to F, that is
the angle *VCp* to the angle *VCP* as 1 to
. And therefore since the angle *VCP*
between the upper and the lower apsis, in an immovable
ellipsis, is of 180 deg. the angle *VCp*
between the same apsides in an orbit which a body
describes with a centripetal force. that is as
will be equal to an angle of
deg. And by the same reasoning
if the centripetal force be as the angle
between the apsides will be found equal to
deg. After the same manner the
problem is solved in more difficult cases. The
quantity to which the centripetal force is proportional.
must always be resolved into a converging series whose denominator is . Then the given
part of the numerator arising from that operation
is to be supposed in the same ratio to that
part of it which is not given, as the given part
of this numerator RGG - RFF + TFF - FFX
is to that part of the same numerator which is
not given. And taking away the superfluous quantities
and writing unity for T, the proportion of
G to F is obtainecx

Cor. 1. Hence I the centripetal force be as any
power of the altitude. that power may be found
from the motion of the apsides; and so contrary-wise.
That is, if the whole angular motion, with
which the body returns to the same apsis, be to
the angular motion of one revolution. or 360 drg.
as any number as *m* to another as *n*, and the altitude
called A; the force will be as the power
of the altitude A; the index of which
power is . This appears by the second examples.
Hence 'tis plain that the force in its recess
from the centre cannot decrease in a greater than a
triplicate ratio of the altitude. A body revolving
with such a force and parting from the apsis, if it
once begins to descend can never arrive at the
lower apsis or least altitude, but will descend to the
centre. describing the curve line treated of in cor.
3. prop. 41. But if it should, at its parting from
the lower apsis begin to ascend never so little, it
will ascend in infinitum and never come to the upper
apsis; but will describe the curve line spoken
of in the lame cot. and cor. 6. prop. 44. So that
where the force in its recess from the centre descreases in a greater than a triplicate ratio of the
altitude, the body at its parting from the apsis, will
either descend to the centre or ascend in infinitum,
according as it descends or ascends at the beginning
of its motion. But if the force in its recess
from the centre either decreases in a less than a triplicate
ratio of the altitude, or increases in any ratio
of the altitude whatsoever; the body will
never descend to the centre, but will at some time
arrive at the lower apsis; and on the contrary, if
the body alternately ascending and descending from
one apsis to another never comes to the centre,
then either the force increases in the recess from
the centre, or it decreases in a less than a triplicate
ratio of the altitude; and the sooner the body
returns from one apsis to another, the farther is
the ratio of the forces from the triplicate ratio.
As if the body should return to and from the
upper apsis by an alternate descent and ascent in 8
revolutions, or in 4, or 2. or 1; that is if an should
be to *n* as 8 or 4 or or 2 or 1, and therefore be , or , or , or
,
then the force will be as , or , or , or
; that is, it will
be reciprocally as , or , or , or ,
If the body after each revolution returns
to the same apsis, and the apsis remains unmoved,
then *m* will be to *n* as 1 to 1,
and therefore will
be equal to or
and therefore the decrease of the forces will be in
a duplicate ratio of the altitude; as was demonstrateds
above. If the body in three fourth parts or two thirds, or one third; or one fourth part
of an entire revolution, return to the same apsis;
*m* will be to *n* as or or or or to 1, and
therefore is equal to
or ,
or , or ; and therefore the force is
either reciprocally as or or directly as
or . Lastly, if the body in its progress
from the upper apsis to the same upper apsis again,
goes over one entire revolution and three deg. more,
and therefore that apsis in each revolution of the
body moves three deg. in consequentia; then *m*
will be to *n* as 363 deg. to 360 deg. or as 121
to 120, and therefore will be equal to
and therefore the centripetal force will
be reciprocally as or reciprocally as
very nearly. Therefore the centripetal
force decreases in a ratio something greater than the
duplicate; but approaching 59 times nearer to
the duplicate than the triplicate.

Cor. 1. Hence also if a body, urged by a centripetal
force which is reciprocally as the square of
the altitude, revolves in an ellipsis whose focus is
in the centre of the forces; and a new and
foreign force should be added to or subducted
from this centripetal force; the motion of the apsides
arising from that foreign force may (by
the third examples) be known; and so on the contrary.
As if the force with which the body revolves
in the ellipsis be as ; and the foreign force subducted as *cA*, and therefore the remaining
force as ; then (by the third exam.) 6
will be equal to 1, *m* equal to 1, and *n* equal to
4; and therefore the angle of revolution between the
apsides is equal to deg. Suppose that
foreign force to be 357.45 parts less than the other
force with which the body revolves in the ellipsis;
that is *c* to be or T being equal to 1,
and then will be 1 or
180.7623, that is, 180 deg. 45 min. 44 sec.
Therefore the body parting from the upper apsis,
will arrive at the lower apsis with an angular motion
of 180 deg. 45 min. 44 sec. and this angular
motion being repeated will return to the upper
apsis; and therefore the upper apsis in each revolution
will go forward 1 deg. 31 m. 28 sec. The
apsis of the Moon is about twice as swift.

So much for the motion of bodies in orbits whose planes pass through the centre of force. It now remains to determine those motions in eccentrical planes. For those authors who treat of the motion of heavy bodies use to consider the ascent and descent of such bodies, not only in a perpendicular direction, but at all degrees obliquity upon any given planes; and for the same reason we are to consider in this place the motions of bodies tending to centres by means of any forces whatsoever. when those bodies move in eccentrical planes. These planes are supposed to be perfectly smooth and polished so as not to retard the motion of the bodies in the least. Moreover in thes demonstrations instead of the planes upon which those bodies roll or sslide, and which are therefore tangent plane; to the bodies, I shall use planes parallel to them, in which the centres of the bodies move, and by that motion describe orbits. And by the same method I afterwards determine the motions of bodies performed in curve superficies.