centripetal force tending to the focus of the ellipsis.
Let S be the focus
of the ellipsis. Draw
SP cutting the diame
ter DK of the ellipsis
in E, and the ordinate
Qv in x ; and com
plete the parallelogram
d.rPR, It is evident
that EP is equal to the
greater semi-axis AC :
for drawing HI froln
the other focus H of
the ellipsis parallel to
EC, because CS, CH
are equal, ES, El will
be also equal ;
so that EP is the half sum of PS, PI, that is (because of
the parallels HI, PR, and the equal angles IPR, HPZ), of PS, PH, which
taken together are equal to the whole axis 2AC. Draw QT perpendicu
lar to SP, and putting L for the princi al latus rectum of the ellipsis (or for
III. OF NATURAL PHILOSOPHY. 117
L X ^R t0 L X Py aS ^R t0 PV that 1S
>
US PE
or AC to PC ; and L X Pv to GvP as L to Gy ; and GvP to Qi>
2 as
to CD- ; and by (Corol. 2, Lem. VII) the points Q, and P coinciding, Qv*
is to Q,r- in the ratio of equality ; and Q,.r
2 or Qv2 is to Q,T2 as EP2 to
PF2
, that is, as CA2 to PF2
, or (by Lem. XII) as CD2 to CB2
. And com
pounding all those ratios together, we shall have L X QR to Q,T2 as AC
X L X PC2 X CD2
, or 2CB2 X PC2 X CD2 to PC X Gv X CD2 X
CB2
, or as 2PC to Gv. But the points Q and P coinciding, 2PC and Gr
are equal. And therefore the quantities L X QR and Q,T2
, proportional
SP2
to these, will be also equal. Let those equals be drawn
into-p^B"?
and L
SP2 X QT2
X SP2 will become equal to--^p . And therefore (by Corol. 1 and
5, Prop. VI) the centripetal force is reciprocally as L X SP2
, that is, re
ciprocally in the duplicate ratio of the distance SP. Q.E.I.
The same otherwise.
Since the force tending to the centre of the ellipsis, by which the body
P may revolve in that ellipsis, is (by Corol. 1, Prop. X.) as the distance
CP of the body from the centre C of the ellipsis ; let CE be drawn paral
lel to the tangent PR of the ellipsis : and the force by which the same body
P may revolve about any other point S of the ellipsis, if CE and PS in-
PE3
tersect in E, will be as ^T3 , (by Cor. 3, Prop. VII.) ; that is, if the point
S is the focus of the ellipsis, and therefore PE be given as SP2
recipro
cally. Q.E.I.
With the same brevity with which we reduced the fifth Problem to the
parabola, and hyperbola, we might do the like here : but because of the
dignity of the Problem and its use in what follows, I shall confirm the other
cases by particular demonstrations.
PROPOSITION XII. PROBLEM VII.
Suppose a body to move in an hyperbola ; it is required to find lite law of
the centripetal force tending to the focus of that figure.
Let CA, CB be the semi-axes of the hyperbola ; PG, KD other con
jugate diameters ; PF a perpendicular to the diameter KD ; and Qv an
ordinate to the diameter GP. Draw SP cutting the diameter DK in E,
and the ordinate Qv in x, and complete the parallelogram QRP.r. It is
evident that EP is equal to the semi-transverse axis AC ; for drawing
HE, from the other focus H of the hyperbola, parallel to EC, because CS,
TH are equal, ES El will be also equal ; so that EP is the half difference
J1S THE MATHEMATICAL PRINCIPLES [Book I
.of PS, PI; that is (be
cause of the parallels IH,
PR, and the equal angles
IPR, HPZ), of PS, PH,
the difference of which is
equal to the whole axis
2AC. Draw Q,T perpen
dicular to SP; and put
ting L for the principal
latus rectum of the hy
perbola (that is, for
2BC2 \ ....
-Tp- )
7 we shall have L
X QR to L X Pv as QR
to Pv, or Px to Pv, that is
(because of the similar tri
angles Pxv, PEC), as PE
to PC, or AC to PC.
And L X Pv will be to
Gv X Pv as L to Gv;
and (by the properties of
the conic sections) the rec
tangle G? P is to Q,v2 as
PC2 to CD2
; and by (Cor. 2, Lem. VII.), Qv2 to Qa* the points Q and P
coinciding, becomes a ratio of equality ; and Q,.r
2 or Qv2
is to Q,T2 as EP2
to PF2
, that is, as CA2 to PF2
, or (by Lem. XII.) as CD2 to CB2
: and,
compounding all those ratios together, we shall have L X Q,R to Q,T2 as
AC X L X PC2 X CD2
, or 2CB2 X PC 2 X CD2 to PC X Gv X CD2
X CB2
, or as 2PC to Gv. But the points P and Q, coinciding. 2PC and
Gv are equal. And therefore the quantities L X Q,R arid Q.T2
, propor
tional to them, will be also equal. Let those equals be drawn into
SP2 sp2 x o/r2
^, and we shall have L X SP2
equal to ^^ . And therefore (by
Cor. 1. and 5, Prop. VI.) the centripetal force is reciprocally as L X SP2
.
hat is, reciprocally in the duplicate ratio of the distance SP. Q,.E.I.
TJie same otherwise.
Find out the force tending from the centre C of the hype rbola. This will
be proportional to the distance CP. But from thence (by Cor. 3, Prop.
PE3
VII.) the force tending to the focus S will be as
-^-^
th; (t is, because PE
is given reciprocally as SP-. Q,.E.I.
SEC. III.] OF NATURAL PHILOSOPHY. 119
And the same way may it be demonstrated, that the body having its cen
tripetal changed into a centrifugal force, will move in the conjugate hy
perbola.
LEMMA XIII.
The latus rectum of a parabola belonging to any vertex is quadruple
the distance of that vertexfrom the focus of thejigurc.
This is demonstrated by the writers on the conic sections.
LEMMA XIV.
Tlie perpendicular, let fallfrom the focus of a parabola on its tangent, is
a mean proportional between the. distances of the focusfrom the poini
of contact, andfrom the principal vertex of the figure.
For, let AP be the parabola, S its
focus, A its principal vertex, P the
point of contact, PO an ordinate to the
principal diameter. PM the tangent
meeting the principal diameter in M.
and SN the perpendicular from the fo-~ M A s o
cus on the tangent : join AN, and because of the equal lines MS and SP,
MN and NP, MA and AC, the right lines AN, OP, will be parallel ; and
thence the triangle SAN will be right-angled at A, and similar to the
equal triangles SNM, SNP j therefore PS is to SN as SN to SA. Q.E.D.
COR. 1. PS2
is to SN2 as PS to SA.
COR. 2. And because SA is given, SN- will be as PS.
COR. 3. And the concourse of any tangent PM, with the right line SN.
drawn from the focus per] endicular on the tangent, falls in the right line
AN that touches the parabola in the principal vertex.
PROPOSITION XIII. PROBLEM VIII.
If a body moves in the perimeter of a parabola ; it is required tofind the.
law of the centripetal force tending to the focus of thatfigure.
Retaining the construction
of the preceding Lemma, let P
be the body in the perimeter
of the parabola ; and from the
place Q,, into which it is next
to succeed, draw QH parallel IS!.
and Q,T perpendicular to SP,
as also Qv parallel to the tan
gent, and mating the diame
ter PG in v, and the distance
120 THE MATHEMATICAL PRINCIPLES [BOOK I.
SP in x. Now. because of the similar triangles Pxv, SPM, and of the
equal sides SP, SM of the one, the sides Px or Q,R and Pv of the other
will be also equal. But (by the conic sections) the square of the ordinate
Q,y is equal to the rectangle under the latus rectum and the segment Pv
of the diameter ;
that is (by Lem. XIII.), to the rectangle 4PS X Pv, or
4PS X Q,R ; and the points P and Q, coinciding, the ratio of Qv to Q,.r
(by Cor. 2, Lem. VII.,) becomes a ratio of equality. And therefore Q,#2
, in
this case, becomes equal to the rectangle 4PS X Q,R. But (because of the
similar triangles Q#T, SPN), Q^2 is to QT2 as PS2 to SN2
, that is (by
Cor. 1, Lem. XIV.), as PS to SA ; that is, as 4PS X QR to 4SA x QR,
and therefore (by Prop. IX. Lib. V., Elem.) QT* and 4SA X QR are
SP2 SP2 X QT2
equal. Multiply these equals by ^-^-,
and ^5 -will become equal
to SP2 X 4SA : and therefore (by Cor. 1 and 5, Prop. VL), the centripetal
force is reciprocally as SP2 X 4SA ; that is, because 4SA is given, recipro
cally in the duplicate ratio of the distance SP. Q.E.I.
COR. 1. From the three last Propositions it follows, that if any body P
goes from the place P with any velocity in the direction of any right line
PR, and at the same time is urged by the action of a centripetal force that
is reciprocally proportional to the square of the distance of the places from
the centre, the body will move in one of the conic sections, having its fo
cus in the centre of force ; and the contrary. For the focus, the point of
contact, and the position of the tangent, being given, a conic section may
be described, which at that point shall have a given curvature. But the
curvature is given from the centripetal force and velocity of the body be
ing given ; and two orbits, mutually touching one the other, cannot be de
scribed by the same centripetal force and the same velocity.
COR. 2. If the velocity with which the body goes from its place P is
such, that in any infinitely small moment of time the lineola PR may be
thereby describe I: and the centripetal force such as in the same time to
move the same body through the space QR ; the body will move in one of
QT2
.
the conic sections, whose principal latus rectum is the quantity Tjfr in its
ultimate state, when thelineoke PR, QR are diminished in infinitum. In
these Corollaries I consider the circle as an ellipsis ; and I except the case
where the body descends to the centre in a right line.
PROPOSITION XIV. THEOREM VI.
Tf several bodies revolve about one common centre, and the centripetal
force is reciprocally in tlie duplicate ratio of the distance of places
from the centre ; I say, that the principal latera recta of tfieir orbits
are in the duplicate ratio of the areas, which the bodies by radii drawn
to the centre describe it\ the same time.
SEC. HI. OF NATURAL PHILOSO1 HY.
For (by Cor 2, Prop. XIII) the latus rectum
QT*. L is equal to the quantity-^-in its ultimate
state when the points P and Q, coincide. But
the lineola QR in a given time is as the gen
erating centripetal force ; that is (by supposi-
QT2
tion), reciprocally as SP2
. And
therefore-^-^
is as Q.T2 X SP2
; that is, the latus rectum L is in the duplicate ratio of
the area QT X SP. Q.E.D.
C?OR. Hence the whole area of the ellipsis, and the rectangle under the
axes, which is proportional to it, is in the ratio compounded of the subduplicate
ratio of the latus rectum, and the ratio of the periodic time. For
the whole area is as the area QT X SP, described in a given time, mul
tiplied by the periodic time.
PROPOSITION XV. THEOREM VII.
The same things being supposed, J say, that the periodic times in ellip
ses are in the sesquiplicate ratio of their greater axes.
For the lesser axis is a mean proportional between the greater axis and
the latus rectum ; and, therefore, the rectangle under the axes is in the
ratio compounded of the subduplicate ratio of the latus rectum and the
sesquiplicate ratio of the greater axis. But this rectangle (by Cor. o.
Prop. XIV) is in a ratio compounded of the subduplicate ratio of the
latus rectum, and the ratio of the periodic time. Subduct from both sides
the subduplicate ratio of the latus rectum, and there will remain the ses
quiplicate ratio of the greater axis, equal to the ratio of the periodic time.
Q.E.D.
COR. Therefore the periodic times in ellipses are the same as in circles
whose diameters are equal to the greater axes of the ellipses.
PROPOSITION XVI. THEOREM VIII.
The same things being supposed, and right lines being drawn to the
bodies that shall touch the orbits, and perpendiculars being let fall on
those tangents from the commonfocus ; I say, that the velocities oj
the bodies are in a ratio compounded of the ratio of the perpendiculars
inversely, and the, subduplicate ratio of the principal latera recta
direct!]).
From the focus S draw SY perpendicular to the tangent PR, and the
velocity of the body P will be reciprocally in the subduplicate ratio of the
SY2
quantity -y . For that velocity is as the infinitely small arc PQ de
122 THE MATHEMATICAL PRINCIPLES [BOOK I.
scribed in a given moment of time, that is (by
Lem. "VII), as the tangent PR ; that is (because
of the proportionals PR to Q,T, and SP to
SP X Q,T
SY), as ~y ; or as SY reciprocally,
and SP X Q,T directly ; but SP X QT is as
the area described in the given time, that is (by
Prop. XIV), in the subduplicate ratio of the
latus rectum. Q.E.D.
COR. 1. The principal latera recta are in a ratio compounded of the
duplicate ratio of the perpendiculars and the duplicate ratio of the ve
locities.
COR. 2. The velocities of bodies, in their greatest and least distances from
the common focus, are in the ratio compounded of the ratio of the distan
ces inversely, and the subduplicate ratio of the principal latera recta di
rectly. For those perpendiculars are now the distances.
COR. 3. Arid therefore the velocity in a conic section, at its greatest or
least distance from the focus, is to the velocity in a circle, at the same dis
tance from the centre, in the subduplicate ratio of the principal latus rec
tum to the double of that distance.
COR. 4. The velocities of the bodies revolving in ellipses, at their mean
distances from the common focus, are the same as those of bodies revolving
in circles, at the same distances ; that is (by Cor. 6. Prop. IV), recipro
cally in the subduplicate ratio of the distances. For the perpendiculars
are now the lesser semi-axes, and these are as mean proportionals between
the distances and the latera recta. Let this ratio inversely be compounded
with the subduplicate ratio of the latera recta directly, and we shall have
the subduplicate ratio of the distance inversely.
COR. 5. In the same figure, or even in different figures, whose principal
latera recta are equal, the velocity of a body is reciprocally as the perpen
dicular let fall from the focus on the tangent.
COR. 6. In a parabola, the velocity is reciprocally in the subduplicate
the distance of the body from the focus of the figure; it is more
variable in the ellipsis, and less in the hyperbola, than according to this
ratio. For (by Cor. 2, Lem. XIV) the perpendicular let fall from the
focus on the tangent of a parabola is in the subduplicate ratio of the dis
tance. In the hyperbola the perpendicular is less variable ; in the ellipsis
more.
COR. 7. In a parabola, the velocity of a body at any distance from the
focus is to the velocity of a body revolving in a circle, at the same distance
from the centre, in the subduplicate ratio of the number 2 to 1 ; in the
ellipsis it is less, and in the hyperbola greater, than according to this ratio,
For (by Cor. 2 of this Prop.) the velocitv at the vertex of a parabola is ir
SEC. III.] OF NATURAL PHILOSOPHY. 123
this ratio, and (by Cor. 6 of this Prop, and Prop. IV) the same proportion
holds in all distances. And hence, also, in a parabola, the velocity is
everywhere equal to the velocity of a body revolving in a circle at half the
distance ;
in the ellipsis it is less, and in the hyperbola greater.
COR. S. The velocity of a body revolving in any conic section is to the
velocity of a body revolving in a circle, at the distance of half the princi
pal latus rectum of the section, as that distance to the perpendicular let
fall from the focus on the tangent of the section. This appears from
Cor. 5.
COR. 9. Wherefore since (by Cor. 6, Prop. IV), the velocity of a body
revolving in this circle is to the velocity of another body revolving in any
other circle reciprocally in the subduplicate ratio of the distances; there
fore, ex czqiiO) the velocity of a body revolving *in a conic section will be
to the velocity of a body revolving* in a circle at the same distance as a
mean proportional between that common distance, and half the principal
latus rectum of the section, to the perpendicular let fall from the common
focus upon the tangent of the section.
PROPOSITION XVII. PROBLEM IX.
Supposing the centripetal force to be reciprocally proportional to the
squares of the distances of places from the centre, and that the abso
lute quantity of that force is known ; it is required to determine t/te
line which a body will describe that is let gofrom a given place with a
given velocity in the direction of a given right line.
Let the centripetal force
tending to the point S be
such as will make the body
p revolve in any given orbit
pq ; and suppose the velocity
of this body in the place p
is known. Then from the
place P suppose the body P
to be let go with a given ve
locity in the direction of the
line PR ; but by virtue of a
centripetal force to be immediately turned aside from that right line into
the conic section PQ,. This, the right line PR will therefore touch in P.
Suppose likewise that the right line pr touches the orbit pq in p ; and if
from S you suppose perpendiculars let fall on those tangents, the principal
latus rectum of the conic section (by Cor. 1, Prop. XVI) will be to the
principal latus rectum of that orbit in a ratio compounded of the duplicate
ratio of the perpendiculars, and the duplicate ratio of the velocities ; arid
is therefore given. Let this latus rectum be L ; the focus S of the conic
L24 THE MATHEMATICAL PRINCIPLES [BOOK I
section is also given. Let the angle RPH be the complement of the angle
RPS to two right ; and the line PH, in which the other focus II is placed,
is given by position. Let fall SK perpendicular on PH, and erect the
conjugate semi-axis BC ; this done, we shall have SP2 2KPH + PH2
= SH2 = 4CH2 = 4BH2 4BC2 = SP + PH2 L X SiM
SP2 + 2SPH + PH2 L x SP + PH. Add on both sides 2KPH
SP2 PH2 + L X SP + PH, and we shall have L X SP + PH=2SPH
f 2KPH, or SP + PH to PH, as 2SP + 2KP to L. Whence PH is
given both in length and position. That is, if the velocity of the body
in P is such that the latus rectum L is less than 2SP + 2KP, PH will
lie on the same side of the tangent PR writh the line SP ; and therefore
the figure will be an ellipsis, which from the given foci S, H, and the
principal axis SP + PH, is given also. But if the velocity of the body
is so great, that the latus rectum L becomes equal to 2SP + 2KP, the
length PH will be infinite
; and therefore, the figure will be a parabola,
which has its axis SH parallel to the line PK, and is thence given. But
if the body goes from its place P with a yet greater velocity, the length
PH is to be taken on the other side the tangent ; and so the tangent pas
sing between the foci, the figure will be an hyperbola having its principal
axis equal to the difference of the lines SP and PH, and thence is given.
Por if the body, in these cases, revolves in a conic section so found, it is
demonstrated in Prop. XI, XII, and XIII, that the centripetal force will
be reciprocally as the square of the distance of the body from the centre
of force S ; and therefore we have rightly determined the line PQ,, which
a body let go from a given place P with a given velocity, and in the di
rection of the right line PR given by position, would describe with such a
force. Q.E.F.
COR. 1. Hence in every conic section, from the principal vertex D, the
latus rectum L, and the focus S given, the other focus H is given, by
taking DH to DS as the latus rectum to the difference between the latus
rectum and 4US. For the proportion, SP + PH to PH as 2SP + 2KP
to L, becomes, in the case of this Corollary, DS + DH to DH as 4DS to
L, and by division DS to DH as 4DS L to L.
COR. 2. Whence if the velocity of a body in the principal vertex D ig
given, the orbit may be readily found ;
to wit, by taking its latus rectum
to twice the distance DS, in the duplicate ratio of this given velocity to
the velocity of a body revolving in a circle at the distance DS (by Cor.
3, Prop. XVI.), and then taking DH to DS as the latus rectum to the
