there be given the circle VR, described
from the centre C with any interval
CV; and from the same centre de
scribe any other circles ID, KE cut
ting the trajectory in I and K, and
the right line CV in D and E. Then
draw the right line CNIX cutting the c
circles KE, VR in N and X, and the right line CKY meeting the circle
VJi in Y. Let the points I and K be indefinitely near ; and let the body
go on from V through I and K to k ; and let the point A be the place
from whence anothe body is to fall, so as in the place D to acquire a ve
locity equal to the velocity of the first body in I. And things remaining
as in Prop. XXXIX, the lineola IK, described in the least given time
THE MATHEMATICAL PRINCIPLES [BOOK 1
trill be as the velocity, and therefore as the right line whose square is
equal to the area ABFD, and the triangle ICK proportional to the time
will be given, and therefore KN will be reciprocally as the altitude 1C :
that is (if there be given any quantity Q, and the altitude 1C be called
A), as -T-. This quantity call Z, and suppose the magnitude of Q, to
oe such that in some case v/ABFD may be to Z as IK to KN, and then
in all cases V ABFD will be to Z as IK to KN, and ABFD to ZZ as
IK2 to KN2
, and by division ABFD ZZ to ZZ as IN2 to KN2
, and therefore
V ABFD ZZ to Z, or as IN to KN; and therefore A x KN
Q. x IN
\vill be equal to . Therefore since YX X XC is to A X KN
ZZ
Q. X IN x CX2
as CX2
, to AA, the rectangle XY X XC will be equal to-
AAv/ABFD ZZ.
Therefore in the perpendicular DF let there be taken continually I)//, IV
Q ax ex2
equal to , =. respectively, and
2 v/ ABFD ZZ 2AA V ABFD ZZ
let the curve lines ab, ac, the foci of the points b and c, be described : and
from the point V let the perpendicular Va be erected to the line AC, cut
ting off the curvilinear areas VD&a, VDra, and let the ordi nates Es:?
E#, be erected also. Then because the rectangle D& X IN or DbzR is
equal to half the rectangle A X KN, or to the triangle ICK ; and the
rectangle DC X IN or Dc.rE is equal to half the rectangle YX X XC, or
to the triangle XCY; that is, because the nascent particles I)6d3, ICK
of the areas VD/>#, VIC are always equal; and the nascent particles
Dc^-E, XCY of the areas VDca, VCX are always equal : therefore the
generated area VD6a will be equal to the generated area VIC, and there
fore proportional to the time; and the generated area VDco- is equal to
the generated sector VCX. If, therefore, any time be given during which
the body has been moving from V, there will be also given the area pro
portional to it VD/>; and thence will be given the altitude of the body
CD or CI ; and the area VDca, and the sector VCX equal there o, together
with its angle VCL But the angb VCI, and the altitude CI being given,
there is also given the place I, in which the body will be found at the end
of that time. Q.E.I.
COR. 1. Hence the greatest and least altitudes of the bodies, that is, the
apsides of the trajectories, may be found very readily. For the apsides
are those points in which a right line 1C drawn through the centre falls
perpendicularly upon the trajectory VTK; which comes to pass when the
right lines IK and NK become equal; that is, when the area ABFD ig
C nl to ZZ.
SEC. VI1LJ OF NATURAL PHILOSOPHY. 171
COR. 2. So also the angle KIN, in which the trajectory at any place
cuts the line 1C. may be readily found by the given altitude 1C of the
body : to wit, by making the sine of that angle to radius as KN to IK
that is, as Z to the square root of the area ABFD.
COR. 3. If to the centre C, and the
principal vertex V, there be described a
conic section VRS ; and from any point
thereof, as R, there be drawn the tangent T
RT meeting the axis CV indefinitely pro
duced in the point T ; and then joining C
CR there be drawn the right line CP, Qequal
to the abscissa CT, making an angle VCP proportional to the sector
VCR ; and if a centripetal force, reciprocally proportional to the cubes
of the distances of the places from the centre, tends to the centre C ; and
from the place V there sets out a body with a just velocity in the direc
tion of a line perpendicular to the right, line CV; that body will proceed
in a trajectory VPQ,, which the point P will always touch ; and therefore
if the conic section VI\ S be an hyberbola, the body will descend to the cen
tre
; but if it be an ellipsis, it will ascend perpetually, and go farther and
farther off in infinilum. And, on the contrary, if a body endued with any
velocity goes off from the place V, and according as it begins either to de
scend obliquely to the centre, or ascends obliquely from it, the figure VRS
be either an hyperbola or an ellipsis, the trajectory may be found by increas
ing or diminishing the angle VCP in a given ratio. And the centripetal
force becoming centrifugal, the body will ascend obliquely in the trajectory
VPQ, which is found by taking the angle VCP proportional to the elliptic
sector VRC, and the length CP equal to the length CT, as before. All these
things follow from the foregoing Proposition, by the quadrature of a certain
ourve, the invention of which, as being easy enough, for brevity s sake I omit.
PROPOSITION XLII. PROBLEM XXIX.
The law of centripetal force being given, it is required to find the motion
of a body setting out from a given place, with a given velocity, in the
direction of a given right line.
Suppose the same things as in
Ihe three preceding propositions;
and let the body go off from
the place I in the direction of the
little line, IK, with the same ve
locity as another body, by falling
with an uniform centripetal force
from the place P, may acquire in
I); and let this uniform force be
to the force with which the body
1.72 THE MATHEMATICAL PRINCIPLES [BOOK 1.
is at first urged in I, as DR to DF. Let the body go on towards k; and
about the centre C, with the interval Ck, describe the circle ke, meeting
the right line PD in e, and let there be erected the lines eg, ev, ew, ordinately
applied to the curves BF*, abv} acw. From the given rectangle
PDRQ, and the given law of centripetal force, by which the first body is
acted on, the curve line BF* is also given, by the construction of Prop.
XXVII, and its Cor. 1. Then from the given angle CIK is given the
proportion of the nascent lines 1K; KN ; and thence, by the construction
of Prob. XXVIII, there is given the quantity Q,, with the curve lines abv,
acw ; and therefore, at the end of any time Dbve, there is given both
the altitude of the body Ce or Ck, and the area Dcwe, with the sector
equal to it XCy, the angle 1CA:, and the place k} in which the body will
then be found. Q.E.I.
We suppose in these Propositions the centripetal force to vary in its
recess from the centre according to some law, which any one may imagine
at pleasure; but at equal distances from the centre to be everywhere the
Bame.
I have hitherto considered the motions of bodies in immovable orbits.
It remains now to add something concerning their motions in orbits which
revolve round the centres of force.
SECTION IX.
Of the motion of bodies in moveable orbits ; and of the motion of the
apsides.
PROPOSITION XLIII. PROBLEM XXX.
Ft is required to make a body move in a trajectory that revolves about
the centre offorce in the same manner as another body in the same
trajectory at rest.
In. the orbit VPK, 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 VC/? 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, ns the velocity of the describing
line Cp to the velocity of the describing line CP ;
that is, as the angle VC/? 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
SEC. L\.] OF NATURAL PHILOSOPHY. 173
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 an immovable plane.
Make the angle VC^ equal to the angle PC/?, and the line Cu equal to
CV, and the figure uCp equal to the figure VCP; and the body being al
ways in the point p} will move in the perimeter of the revolving figure
nCp, and will describe its (revolving) arc up in the same time the* the
other body P describes the similar and equal arc VP in the quiescov.t fig
ure YPK. Find, then, by Cor. 5, Prop. VI., the centripetal force by which
the body may be made to revolve in the curve line which the pom* p de
scribes in an immovable plane, and the Problem will be solved. O/E.K.
PROPOSITION XLIV. THEOREM XIV.
The difference of the forces, by which two bodies may be madi, to KMVG
equally, one in a quiescent, the other in the same orbit revolving, i 1 in
a triplicate ratio of their common altitudes inversely.
Let the parts of the quiescent or
bit VP, PK 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 VC/? to the /2\-
angle VCP. Because the altitudes
of the bodies PC and pV, KG 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
centre, 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 mo
tions 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 mo
tion of the line pC to the angular motion of the line PC ; that is, as the
angle VC/? to the angle VCP. Therefore, at the same time that the bodv
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 ;
arid
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
174 THE MATHEMATICAL PRINCIPLES [BOOK J.
C, 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 VC/? to
the angle VCP, that is, as the transverse motion of the body p to the
transverse motion of the body P, it is manifest that the body p, at the ex
piration of that time, will be found in the place m. These things will be
so, if the bodies jo and P are equally moved in the directions of the lines
pC and PC, and are therefore urged with equal forces in those directions.
I: ut if we take an angle pCn that is to the angle pCk as the angle VGj0
to the angle VCP, and nC be equal to kG, 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 npk, move either in cmiseqnentia, or in antecedenticij
with a celerity greater than the double of that with which the line
CP moves in conseqnentia ; and with a less force if the orbit moves slower
in antecedent-la. And ihj 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, tun pro
duced in s and , and the rectangle mn X nit will be equal to the rectan-
*//? n ^* */?? ^
"le mk X ins, and therefore mn will be equal to . But since
mt
the triangles pCk, pCn, in a given time, are of a given magnitude, kr and
mr. a id their difference mk, and their sum ms, are reciprocally as the al
titude pC, and therefore the rectangle mk X ms is reciprocally as the
square of the altitude pC. But, moreover, mt is directly as |//z/, that is, as
the altitude pC. These are the first ratios of the nascent lines ; and hence
r - that is, the nascent lineola mn. and the difference of the forces
mt
proportional thereto, are reciprocally as the cube of the altitude pC.
Q.E.D.
COR. I. Hence the difference of the forces in the places P and p, or K and
/.*, 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 de
scribes the arc PK, as the nascent line m,n to the versed sine of the nascent
mk X ms rk2
arc RK, that is, as to ^g, 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 VQ?, as GG FF to FF.
And, therefore, 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
OEC. IX.l OF NATURAL PHILOSOPHY. 175
revolving in an immovable orbit has by a radius drawn to the centre debribed
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 movable or
bit, 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 GG FF to FF. For that sector and
the area pCk are to one another as the times in which they are described.
COR. 2. If the orbit YPK be an
ellipsis, having its focus C, and its
highest apsis Y, and we suppose the
the ellipsis upk similar and equal to ..
it, so that pC may be always equal /
to PC, and the angle YC/? be to the ;
angle YCP in the given ratio of G \
to F ; and for the altitude PC or pC \
we put A, and 2R for the latus rec- /t\
turn of the ellipsis, the force with *
which a body may be made to re
volve in a movable ellipsis will be as
FF RGG RFF
- + -
-rg , and vice versa.
/Y A. A.
Let the force with which a body may
revolve in an immovable ellipsis be expressed by the quantity , and the
-. 7
force in V will be
FF
But the force with which a body may revolve in
a circle at the distance CY, with the same velocity as a body revolving in
an ellipsis has in Y, is to the force with which a body revolving in an ellip
sis is acted upon in the apsis Y, as half the latus rectum of the ellipsis to the
RFF
semi-diameter CY of the circle, and therefore is as , =- : and tlu
RFF
which is to this, as GG FF to FF, is as -
~py^~~
~
: and this force
(by Cor. 1 cf this Prop.) is the difference of the forces in Y, with which the
body P revolves in the immovable ellipsis YPK, and the body p in the
movable ellipsis upk. Therefore since by this Prop, that difference at
any other altitude A is to itself at the altitude CY as -r-, to ^TF- the same AJ CYJ
R C^ ("* R P^ T*
1
difference in every altitude A will be as -
3
:
. Therefore to the
FF
force -T-:
, by which the body may revolve in an immovable ellipsis VPK
176 THE MATHEMATICAL PRINCIPLES [BOOK I.
idd the excess -:-=
A , and the sum will be the whole force A-rA-r -\-
RGG RFF,
.-5 by which a body may revolve in the same time in the mot-
A.
able ellipsis upk.
COR. 3. In the same manner it will be found, that, if the immovable or
bit VPK be an ellipsis having its centre in the centre of the forces C} and
there be supposed a movable ellipsis -upk, similar, equal, and concentrical
to it
; and 2R be the principal latus rectum of that ellipsis, and 2T the
latus transversum, or greater axis ; and the angle VCjo be continually to the
angle TCP as G to F ; the forces with which bodies may revolve in the im-
FFA FFA
movable and movable ellipsis, in equal times, will be as ^ and -p~
RGG RFF
+ A.-3 respectively.
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 Y, 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
VFF
V be called -
f-=Trri , and in other places P be indefinitely styled X ; and the
altitude CP be called A, and G be taken to F in the given ratio of the
angle VCjD 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
upk revolving with a circular motion, will be as the sum of the forces X -f-
VRGG VRFF
~
A*
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 the line VP of an indeterminate
-p length, perpendicular to the line CV given by po
sition, and CP be drawn, and Cp equal to it, mak
ing the angle VC/? having a given ratio to the an
gle VCP, the force with which a body may revolve
in the curve line Vjo/r, which the point p is con
tinually describing, will be reciprocally as the cube
C
of the altitude Cp. For the body P, by its vis in
ertia 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
SEC. IX..J OF NATURAL PHILOSOPHY. 177
body will deflect from the rectilinear motion into the curve line Ypk. But
this curve ~Vpk is the same with the curve VPQ found in Cor. 3, Prop
XLI, in which, I said, hodies attracted with such forces would ascend
obliquely.
PROPOSITION XLV. PROBLEM XXXL
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 movable 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 any other
altitude CP or C/?, and X for the difference of the altitudes CV CP :
and the force writh which a body moves in an ellipsis revolving about its
p-p T? C* f^
T? F*F
focus C (as in Cor. 2), and which in Cor. 2 was as -r-r -\ -.-3 ,
FFA + RGG RFF ,
that is as, -^ , by substituting T X for A, will be- A
RGG RFF + TFF FFX
come as
-p . In like manner any other cen
tripetal force is to be reduced to a fraction whose denominator is A3
, and
the numerators are to be made analogous by collating together the homo
logous terms. This will be made plainer by Examples.
EXAMPLE 1. Let us suppose the centripetal force to be uniform,
A3
and therefore as 3 or, writing T X for A in the numerator, as
T3 3TTX + 3TXX X3
=-. Ihen collating together the correspon- A3
dent terms of the numerators, that is, those that consist of given quantities,
with those of given quantities, and those of quantities not given with those
of quantities not given, it will become RGG RFF -f- TFF to T3 as
FFX to 3TTX -f 3TXX X3
, or as FF to 3TT + 3TX XX.
Now since the orbit is supposed extremely 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 ratios will be, as RGG to T2
, 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 VC/?
to the angle VCP, as 1 to v/3. Therefore since the body, in an immovable
178 THE MATHEMATICAL PRINCIPLES [BOOK I
ellipsis, in descending from the upper to the lower apsis, describes an angle,
if I may so speak, of ISO deg., the other body in a movable ellipsis, and there
fore in the immovable orbit we are treating of, will in its descent from
180
the upper to the lower apsis, describe an angle VCjt? of ^ deg. And this
\/o
comes to pass 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
