COR. 6. If the periodic times are in the sesquiplicate ratio of the radii,
and therefore the velocities reciprocally in the subduplicate ratio of the
radii, the centripetal forces will be in the duplicate ratio of the radii in
versely : and the contrary.
COR. 7. And universally, if the periodic time is as any power Rn of the
radius R, and therefore the velocity reciprocally as the power Rn ] of
the radius, the centripetal force will be reciprocally as the power R2n 1 of
the radius; and the contrary.
COR. 8. The same things all hold concerning the times, the velocities,
and forces by which bodies describe the similar parts of any similar figures
that have their centres in a similar position with those figures ; as appears
by applying the demonstration of the preceding cases to those. And the
application is easy, by only substituting the equable description of areas in
the place of equable motion, and using the distances of the bodies from the
centres instead of the radii.
COR. 9. From the same demonstration it likewise follows, that the arc
which a body, uniformly revolving in a circle by means of a given centri
petal force, describes in any time, is a mean proportional between the
diameter of the circle, and the space which the same body falling by the
same given force would descend through in the same given time.
SCHOLIUM.
The case of the 6th Corollary obtains in the celestial bodies (as Sir
Christopher Wren, Dr. Hooke, and Dr. Halley have severally observed) ;
and therefore in what follows, I intend to treat more at large of those
things which relate to centripetal force decreasing in a duplicate ratio
of the distances from the centres.
Moreover, by means of the preceding Proposition and its Corollaries, we
SEC. II.] OF NATURAL PHILOSOPHY. 109
may discover the proportion of a centripetal force to any other known
force, such as that of gravity. For if a body by means of its gravity re
volves in a circle concentric to the earth, this gravity is the centripetal
force of that body. But from the descent of heavy bodies, the time of one
entire revolution, as well as the arc described in any given time, is given
(by Cor. 9 of this Prop.). And by such propositions, Mr. Huygens, in his
excellent book De Horologio Oscillatorio, has compared the force of
gravity with the centrifugal forces of revolving bodies.
The preceding Proposition may be likewise demonstrated after this
manner. In any circle suppose a polygon to be inscribed of any number
of sides. And if a body, moved with a given velocity along the sides of the
polygon, is reflected from the circle at the several angular points, the force,
with which at every reflection it strikes the circle, will be as its velocity :
and therefore the sum of the forces, in a given time, will be as that ve
locity and the number of reflections conjunctly ; that is (if the species of
the polygon be given), as the length described in that given time, and in
creased or diminished in the ratio of the same length to the radius of the
circle ; that is, as the square of that length applied to the radius ; and
therefore the polygon, by having its sides diminished in inftnitum, coin
cides with the circle, as the square of the arc described in a given time ap
plied to the radius. This is the centrifugal force, with which the body
impels the circle
; and to which the contrary force, wherewith the circle
continually repels the body towards the centre, is equal.
PROPOSITION V. PROBLEM I.
There being given, in any places, the velocity with which a body de
scribes a given figure, by means of forces directed to some common
centre : to find that centre.
Let the three right lines PT, TQV, VR
touch the figure described in as many points,
P, Q, R, and meet in T and V. On the tan
gents erect the perpendiculars PA, QB, RC,
reciprocally proportional to the velocities of the
body in the points P, Q, R, from which the
perpendiculars were raised ; that is, so that PA
may be to QB as the velocity in Q to the velocity in P, and QB to RC
as the velocity in R to the velocity in Q. Through the ends A, B, C, of
the perpendiculars draw AD, DBE, EC, at right angles, meeting in D and
E : and the right lines TD, VE produced, will meet in S, the centre re
quired.
For the perpendiculars let fall from the centre S on the tangents PT.
QT. are reciprocally as the velocities of the bodies in the points P and Q
110 THE MATHEMATICAL PRINCIPLES [BOOK 1
(by Cor. 1, Prop. I.), and therefore, by construction, as the perpendiculars
AP, BQ, directly ;
that is, as the perpendiculars let fall from the point D
on the tangents. Whence it is easy to infer that the points S, D, T, are
in one right line. And by the like argument the points S, E, V are also
in one right line ; and therefore the centre S is in the point where the
right lines TD; YE meet. Q.E.D.
PROPOSITION VL THEOREM V.
In a space void of resistance, if a body revolves in any orbit about an im
movable centre, and in the least time describes any arc just then, na
scent ; and the versed sine of that arc is supposed to be drawn bisect
ing the chord, and produced passing through the centre offorce: the
centripetal force in the middle of the arc will be as the versed sine di
rectly and the square of the time inversely.
For the versed sine in a given time is as the force (by Cor. 4, Prop. 1) ;
and augmenting the time in any ratio, because the arc will be augmented
in the same ratio, the versed sine will be augmented in the duplicate of
that ratio (by Cor. 2 and 3, Lem. XL), and therefore is as the force and the
square of the time. Subduct on both sides the duplicate ratio of the
time, and the force will be as the versed sine directly, arid the square of
the time inversely. Q.E.D.
And the same thing may also be easily demonstrated by Corol. 4 ?
T,em. X.
COR. 1. If a body P revolving about the
centre S describes a curve line APQ,, which a
right line ZPR touches in any point P ; and
from any other point Q, of the curve, QJl is
drawn parallel to the distance SP, meeting
the tangent in R ; and QT is drawn perpen-
(licular to the distance SP ; the centripetal force will be reciprocally as the
sp2 x Q/r2
solid- :
, if the solid be taken of that magnitude which it ultimately
acquires when the points P and Q, coincide. For Q,R is equal to
the versed sine of double the arc QP, whose middle is P : and double the
triangle SQP, or SP X Q,T is proportional to the time in which that
double arc is described ; and therefore may be used for the exponent of
the time.
COR. 2. By a like reasoning, the centripetal force is reciprocally as the
SY2 X QJP2
solid-7^5-;
if SY is a perpendicular from the centre of force on
PR the tangent of the orbit. For the rectangles SY X QP and SP X Q,T
are equal.
SEC. II.] OF NATURAL PHILOSOPHY. Ill
COR. 3. If the orbit is cither a circle, or touches or cuts a circle c< ncentrically,
that is, contains with a circle the least angle of contact or sec
tion, having the same curvature rnd the same radius of curvature at the
point P : and if PV be a chord of this circle, drawn from the body through
the centre of force ;
the centripetal force will be reciprocally as the solid
QP2
SY2 X PV. For PV is -.
COR. 4. The same things being supposed, the centripetal force is as the
square of the velocity directly, and that chord inversely. For the velocity
is reciprocally as the perpendicular SY, by Cor. 1. Prop. I.
COR. 5. Hence if any curvilinear figure APQ, is given, and therein a
point S is also given, to which a centripetal force is perpetually directed.
that law of centripetal force may be found, by which the body P will bcj
continually drawn back from a rectilinear course, and. being detained in
the perimeter of that figure, will describe the same by a perpetual revolu-
SP2 x QT2
tion. That is, we are to find, by computation, either the solid -----
or the solid SY2 X PV, reciprocally proportional to this force. Example:
of this we shall give in the following Problems.
PROPOSITION VII. PROBLEM II.
Tf a body revolves in the circumference of a circle; it is proposed to finii
the law of centripetal force directed to any given, point.
Let VQPA be the circumference of the
circle ; S the given point to which as to
a centre the force tends : P the body mov
ing in the circumference ; Q the next
place into which it is to move; and PRZ
the tangent of the circle at the preceding
place. Through the point S draw the
v
chord PV, and the diameter VA of the
circle : join AP, and draw Q,T perpen
dicular to SP, which produced, may meet
the tangent PR in Z ; and lastly, through
the point Q, draw LR parallel to SP, meeting the circle" in L, and the
tangent PZ in R. And, because of the similar triangles ZQR, ZTP.
VPA, we shall have RP2
, that is. QRL to QT2 as AV2 to PV2
. And
QRlj x PV2 SI3 -
therefore - TS--is equal to QT2
. Multiply those equals by -
.
and the points P and Q, coinciding, for RL write PV ; then we shall have
SP- X PV5 SP2 x QT2
And therefore flr Cor 1 and 5. Prop. VI.)
112 THE MATHEMATICAL PRINCIPLES [BOOK I,
SP2 X PV3
the centripetal force is reciprocally as -
ry^~ J that is (because AV2
ia given), reciprocally as the square of the distance or altitude SP, and the
3ube of the chord PV conjunctly. Q.E.L
The same otherwise.
On the tangent PR produced let fall the perpendicular SY ; and (be
cause of the similar triangles SYP, VPA), we shall have AV to PV as SP
SP X PV SP2
>< PV3
to SY, and therefore--^~
- = SY, and - ^- = SY2 A V A X PV. V
And therefore (by Corol. 3 and 5, Prop. VI), the centripetal force is recip-
SP2 X PV3
rocally as -
~~ry~~~ I
*na* *s (because AV is given), reciprocally as SP"
X PV3
. Q.E.I.
Con. 1. Hence if the given point S, to which the centripetal force al
ways tends, is placed in the circumference of the circle, as at V, the cen
tripetal force will be reciprocally as the quadrato-cube (or fifth power) of
the altitude SP.
COR. 2. The force by which the body P in the
circle APTV revolves about the centre of force S
is to the force by which the same body P may re
volve in the same circle, and in the same periodic
time, about any other centre of force R, as RP2 X
SP to the cube of the right line SG, which, from
the first centre of force S is drawn parallel to the
distance PR of the body from the second centre of force R, meeting the
tangent PG of the orbit in G. For by the construction of this Proposition,
the former force is to the latter as RP2 X PT3 to SP2 X PV3
; that is, as
SP3 X PV3
SP X RP2 to --
p ; or (because of the similar triangles PSG, TPV)
to SGS
.
COR. 3. The force by which the body P in any orbit revolves about the
centre of force S, is to the force by which the same body may revolve in
the same orbit, and the same periodic time, about any other centre of force
R. as the solid SP X RP2
, contained under the distance of the body from
the first centre of force S, and the square of its distance from the sec
ond centre of force R, to the cube of the right line SG, drawn from the
first centre of the force S, parallel to the distance RP of the body from
fch*3 second centre of force R, meeting the tangent PG of the orbit in G.
For the force in this orbit at any point P is the same as in a circle of the
same curvature.
SJSG. IL] OF NATURAL PHILOSOPHY. 113
PROPOSITION VIII. PROBLEM III.
If a body mi ues in the semi-circuwferencePQA: it is proposed to find
the law of the centripetal force tending to a point S, so remote, that all
the lines PS. RS drawn thereto, may be taken for parallels.
From C, the centre of the semi-circle, let
the semi-diameter CA he drawn, cutting the
parallels at right angles in M and N, and
join CP. Because of the similar triangles
CPM, PZT, and RZQ, we shall have CP2
to PM2 as PR2 to QT2
; and, from the na
ture of the circle, PR2
is equal to the rect
angle QR X RN + QN, or, the points P, Q coinciding, to the rectangle
QR x 2PM. Therefore CP2
is to PM2 as QR X 2PM to QT2
; and
QT2 2PM3 QT2 X SP2 2PM3 X SP2
QR therefore (by
Corol.
8PM3 X SP2
, and QR And
1 and 5, Prop. VI.), the centripetal force is reciprocally as
2SP2
.
that is (neglecting the given ratio
-ppr)> reciprocally as
PM3
. Q.E.L
And the same thing is likewise easily inferred from the preceding Pro
position.
SCHOLIUM.
And by a like reasoning, a body will be moved in an ellipsis, or even ia
an hyperbola, or parabola, by a centripetal force which is reciprocally ae
the cube of the ordinate directed to an infinitely remote centre of force.
PROPOSITION IX. PROBLEM IV.
If a body revolves in a spiral PQS, cutting all the radii SP, SQ, fyc.,
in a given angle; it is proposed to find thelaio of the centripetal force
tending to tJie centre of that spiral.
Suppose the inde
finitely small angle AY
PSQ to be given ; be
cause, then, all the
angles are given, the
figure SPRQT will ,_ be given in specie.
v
QT Q,T2
Therefore the ratio-7^- is also given, and is as QT, that is (be
lot IX QK
cause the figure is given in specie), as SP. But if the angle PSQ is any
way changed, the right line QR, subtending the angle of contact QPU
tU THE MATHEMATICAL PRINCIPLES [BOOK J
(by Lemma XI) will be changed in the duplicate ratio of PR or QT
QT2
Therefore the ratio ~TVD~remains the same as before, that is, as SP. And
QT2 x SP2
-^ is as SP3
, and therefore (by Corol. 1 and 5, Prop. YI) the
centripetal force is reciprocally as the cube of the distance SP. Q.E.I.
The same otherwise.
The perpendicular SY let fall upon the tangent, and the chord PY of
the circle concentrically cutting the spiral, are in given ratios to the height
SP ; and therefore SP3
is as SY2 X PY, that is (by Corol. 3 and 5, Prop.
YI) reciprocally as the centripetal force.
LEMMA XII.
All parallelograms circumscribed about any conjugate diameters of a
given ellipsis or hyperbola are equal among themselves.
This is demonstrated by the writers on the conic sections.
PROPOSITION X. PROBLEM Y.
If a body revolves in an ellipsis ; it is proposed to find the law of thi
centripetal force tending to the centre of the ellipsis.
Suppose CA, CB to
be semi-axes of the
ellipsis; GP, DK, con
jugate diameters ; PF,
Q,T perpendiculars to
those diameters; Qvan
^rdinate to the diame
ter GP ; and if the
parallelogram QvPR
be completed, then (by
the properties of the
jonic sections) the reclangle
PvG will be to
Qv2 as PC2 to CD2
;
and (because of the
similar triangles Q^T, PCF), Qi>
2 to QT2 as PC2 to PF2
; and, by com
position, the ratio of PvG to QT2
is compounded of the ratio of PC2
1<
QT2
CD2
, and of the ratio of PC2 to PF2
, that is, vG to
-p
as PC;
to_92L^_
P
_
]
^_. Put QR for Pr, and (by Lem. XII) BC X CA for CD
K PF ; also (the points P and Q coinciding) 2PC for rG; and multiply
SEC. II.] OF NATURAL PHILOSOPHY. 115
QT2 x PC2
ing the extremes and means together, we shall have rfo~ equal to
2BC2 X CA2
pp . Therefore (by Cor. 5, Prop. VI), the centripetal force is
2BC2 X CA2
reciprocally as ry~ ; that is (because 2I3C2 X CA2
is given), re
ciprocally as-r^v; that is, directly as the distance PC. QEI.
I O
TJie same otherwise.
[n the right line PG on the other side of the point T, take the point u
so that Tu may be equal to TV ; then take uV, such as shall be to vG as
DC2 to PC2
. And because Qr9
is to PvG as DC2 to PC2
(by the conic
sections), we shall have Qv2 -= Pi X V. Add the rectangle n.Pv to both
sides, and the square of the chord of the arc PQ, will be equal to the rect
angle VPv ; and therefore a circle which touches the conic section in P,
and passes through the point Q,, will pass also through the point V. Now
let the points P and Q, meet, and the ratio of nV to rG, which is the same
with the ratio of DC2 to PC2
, will become the ratio of PV to PG, or PV
2DC2
to 2PC : and therefore PY will be equal to . And therefore the
force by which the body P revolves in the ellipsis will be reciprocally as
2 DC2
ry X PF2
(by Cor. 3, Prop. VI) ; that is (because 2DC2 X PF2
is
I O
given) directly as PC. Q.E.I.
COR. 1. And therefore the force is as the distance of the body from the
centre of the ellipsis ; and, vice versa, if the force is as the distance, the
body will move in an ellipsis whose centre coincides with the centre of force,
or perhaps in a circle into which the ellipsis may degenerate.
COR. 2. And the periodic times of the revolutions made in all ellipses
whatsoever about the same centre will be equal. For those times in sim
ilar ellipses will be equal (by Corol. 3 and S, Prop. IV) ; but in ellipses
that have their greater axis common, they are one to another as the whole
areas of the ellipses directly, and the parts of the areas described in the
same time inversely: that is, as the lesser axes directly, and the velocities
of the bodies in their principal vertices inversely ; :hat is, as those lesser
axes dirtily, and the ordinates to the same point
% f the common axes in
versely ; and therefore (because of the equality of the direct and inverse
ratios) in the ratio of equality.
SCHOLIUM.
If the ellipsis, by having its centre removed to an infinite distance, de
generates into a parabola, the body will move in tin s parabola ; and the
116 THE MATHEMATICAL PRINCIPLES [BOOK I
force, now tending to a centre infinitely remote, will become equable.
Which is Galileo s theorem. And if the parabolic section of the cone (by
changing the inclination of the cutting plane to the cone) degenerates into
an hyperbola, the body will move in the perimeter of this hyperbola, hav
ing its centripetal force changed into a centrifugal force. And in like
manner as in the circle, or in the ellipsis, if the forces are directed to the
centre of the figure placed in the abscissa, those forces by increasing or di
minishing the ordinates in any given ratio, or even by changing the angle
of the inclination of the ordinates to the abscissa, are always augmented
or diminished in the ratio of the distances from the centre ; provided the
periodic times remain equal ; so also in all figures whatsoever, if the ordinates
are augmented or diminished in any given ratio, or their inclination
is any way changed, the periodic time remaining the same, the forces di
rected to any centre placed in the abscissa are in the several ordinatee
augmented or diminished in the ratio of the distances from the centre
SECTION III.
Of the motion of bodies in eccentric conic sections.
PROPOSITION XL PROBLEM VI.
If a body revolves in an ellipsis ; it is required to find the law of the
