body performing its circuits in a revolving ellipsis will describe in a quies
cent 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 figure. A body, therefore revolving with an uniform centripetal
180
force in an orbit nearly circular, will always describe an angle of deg/, or
v/o
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 infinitwn.
EXAM. 2. Suppose the centripetal force to be as any power of the alti-
An
tude A, as, for example, An 3
, or-r^ ; where n 3 and n signify any in-
A.
dices of powers whatever, whether integers or fractions, rational or surd,
affirmative or negative. That numerator An or T X| n
being reduced to
an indeterminate series by my method of converging series, will become
Tn
>/XT
n T + ^XXTn 2
, &c. And conferring these terms
with the terms of the other numerator RGG RFF + TFF FFX, it
becomes as RGG RFF 4- TFF to Tn
, so FF to ?/.T
n r + ?~^
XTn 2
, &c. And taking the last ratios where the orbits approach to
circles, it becomes as RGG to T 1
, so FF to nT-1 T
, or as GG to
T"
, so FF to ?*Tn
; and again, GG to FF, so Tn l to nT"
1
, that
is, as 1 to n ; and therefore G is to F, that is the angle VCp to the angle
VCP, as 1 to ^/n. Therefore since the angle VCP, described in the de
scent of the body from the upper apsis to the lower apsis in an ellipsis, is
of 180 deg., the angle VC/?, described in the descent of the body from the
upper apsis to the lower apsis in an orbit nearly circular which a body de
scribes with a centripetal force proportional to the power An 3
, will be equal
ISO
to an angle of -
deg., and this angle being repeated, the body will re-
\/ti
turn 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,
A4
or -p, n will be equal to 4, and ^/n equal to 2 ; and thereLre the angle
IX.] OF NATURAL PHILOSOPHY. IT9
ISO
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 infiuitum. This appears
also from Prop. X. For a body acted on by this centripetal force will re
volve in an immovable ellipsis, whose centre is the centre of force. If the
1 A 2
centripetal force is reciprocally as the distance, that is, directly as or A A"
ji will be equal to 2 ; and therefore the angle between the upper and lower
180
apsis will be -
deg., or 127 deg., 16 min., 45 sec. ; and therefore a body re
v/2
volving 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 A , and, therefore, directly as -r-fp or as
Ts>
n wil* ^e et
l ual f
\>
an(1
4 A^- A
1 Of)
-
deg. will be equal to 360 deg. ; and therefore the body parting from
v/ n
the upper apsis, and from thence perpetually descending, will arrive at the
lower apsis when it has completed one entire revolution ; and thence as
cending perpetually, when it has completed another entire revolution, it
will arrive again at the upper apsis ; and so alternately for ever.
EXAM. 3. Taking m and n for any indices of the powers of the alti
tude, and b and c for any given numbers, suppose the centripetal force
6Ara + cA" b into T X> -f- c into T X
to be as r^ that is, as A3 A3
or (by the method of converging series above-mentioned) as
bTm + cTn
m6XT" - 1 //cXTn mm m vvrpm un n
~~2
--0A.A1 ^
t-XXT"
2
, fcc.
T$~
~ and comparing the terms of the numerators, there will
arise RGG IIFF -f TFF to ^Tm + cT" as FF to mbTm i
" - + 2
" m
bXT" - * + "^p
cXTn - .fee. And taking
the last ratios that arise when the orbits come to a circular form, there
will come forth GG to 6Tm l -f cTn 1 as FF to mbTm l + ncT"
J
;
and again, GG to FF as 6Tm + cTn to mbTn 1
-f ncTn \
This proportion, by expressing the greatest altitude CV or T arithmeti
cally by unity, becomes, GG to FF as b -{- c to mb -\- ?/c, and therefore as I
(80 THE MATHEMATICAL PRINCIPLES [BOOK 1
tub ~h nc
to -
y7 Whence G becomes to P, that is, the angle VCjo to the anf)
~T~ C
gle 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., thr
angle VC/? between the same apsides in an orbit which a body describes
b A m I c A n
with a centripetal force, that is. as - r , will be equal to an angle of
A.
ISO v/ 1~TT~; deg. And y tne same reasoning, if the centripetal force
be as -
73 , the angle between the apsides will be found equal to
fi f*
18o V - -
deg. After the same manner the Problem is solved in
nib >ic
more difficult cases. The quantity to which the centripetal force is pro
portional must always be resolved into a converging series whose denomi
nator is A*. 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 -f 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 obtained.
COR. 1 . Hence if the centripetal force be as any power of the altitude,
that power may be found from the motion of the apsides ; and so contra
riwise. 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 deg.,
MS any number as m to another as n, and the altitude called A ; the force
nn
will be as the power A HSii
3 of the altitude A; the index of which power is
- 3. This appears by the second example. Hence it is 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. XLL But if it should, at its part
ing from the lower apsis, begin to ascend never so little, it will ascend in
irtfimtifm, and never come to the upper apsis ;
but will describe the curve
line spoken of in the same Cor., and Cor. 6, Prop. XLIV. So that where
the force in its recess from the centre decreases in a greater than a tripli
cate ratio of the altitude, the body at its parting from the apsis, will either
descend to the centre, or ascend in iiiftnitum, according as it descends or
Ascends at the beginning of its motion. But if the force in its recess from
"SEC. IX.J OF NATURAL PHILOSOPHY. ISi
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 \\ that is, if m should be to n as 8,
or 4, or 2, or H to 1. and therefore ---3, be g\ 3,or TV~3, or i
mm 3, or
3
I - 3
; then the force will be as A~ ? or AT "~
3j or A*~~ 3j or A""
G
that is. it will be reciprocally as A 3 C4 or A 3 T ^ or A 3
4 or A 3
""
If the body after each revolution returns to the same apsis, and the apsis
nn _
remains unmoved, then m will be to n as 1 to 1, and therefore A"
will be equal to A 2
, or -
; and therefore the decrease of the forces will AA
be in a duplicate ratio of the altitude ; as was demonstrated 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 ?
n n i_6 _ 3
9 _ 3
o
or ^ or i to 1, and therefore Amm 3 is equal to A 9 or A4 or A
_ 3 1 6 _ 3 l_l
or A ; and therefore the force is either reciprocally as A fl or
3 613
A 4 or directly as A or A . 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 u as 363 deg. to
360 deg. or as 121 to 120, and therefore Amm will be equal to
2 9_ 5_ 2_ JJ A "
and therefore the centripetal force will be reciprocally as
^T4"6TT> or recip rocally as A 2 ^ 4 ^
very nearly. Therefore the centripetal
force decreases in a ratio something greater than the duplicate ; but ap
proaching 59f times nearer to the duplicate than the triplicate.
COR. 2. Hence also if a body, urged by a centripetal force which is re
ciprocally 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 Example) be known ; and so on
the contrary. As if the force with which the body revolves in the ellipsis
182 THE MATHEMATICAL PRINCIPLES [BOOK I
oe as -r-r-
A ; and the foreign force subducted as cA, and therefore the remain- .A.
^ c^4
ing force as
-^ ; then (by the third Example) b will be equal to 1.
m equal to 1, and n equal to 4 ; and therefore the angle of revolution be
1 c
tween the apsides is equal to 180 <*/- 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 -3 }y j ; A or T being equal to 1
; and then
l8(Vl~4c
will be 18<Vfff Jf 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 min.,
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 eccen
trical planes. For those authors who treat of the motion of heavy bodies
used to consider the ascent and descent of such bodies, not only in a per
pendicular direction, but at all degrees of 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 these demonstrations, instead of the planes upon
which those bodies roll or slide, and which are therefore tangent planes 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 performer
1 in curve superficies.
SECTION X.
Of the motion of bodies in given superficies, and of the reciprocal motion
offnnependulous bodies.
PROPOSITION XLVI. PROBLEM XXXII.
Any kind of centripetal force being supposed, and the centre offorce,
atfft
any plane whatsoever in which the body revolves, being given, and tint
quadratures of curvilinear figures being allowed; it is required to de
termine the motion of a body going off from a given place., with a
given velocity, in the direction of a given right line in, that plane.
SEC. X.J OF NATURAL PHILOSOPHY- 183
Let S be the centre of force, SC the
least distance of that centre from the given
plane, P a body issuing from the place P
in the direction of the right line PZ, Q,
the same body revolving in its trajectory,
and PQ,R the trajectory itself which is
required to be found, described in that
given plane. Join CQ, Q.S, and if in Q,S
we take SV proportional to the centripetal
force with which the body is attracted to
wards the centre S, and draw VT parallel
to CQ, and meeting SC in T ; then will the force SV be resolved into
two (by Cor. 2, of the Laws of Motion), the force ST, and the force TV ;
of
which ST aMracting the body in the direction of a line perpendicular to
that plane, does not at all change its motion in that plane. But the action
c f the other force TV, coinciding with the position of the plane itself, at
tracts the body directly towards the given point C in that plane ; ad
t icreftre causes the body to move in this plane in the same manner as if
the force S F were taken away, and the body were to revolve in free space
about the centre C by means of the force TV alone. But there being given
the centripetal force TV with which the body Q, revolves in free space
about the given centre C, there is given (by Prop. XLII) the trajectory
PQ.R which the body describes ; the place Q,, in which the body will be
found at any given time ; and, lastly, the velocity of the body in that place
Q,. And so e contra. Q..E.I.
PROPOSITION XLV1L THEOREM XV.
Supposing the centripetal force to be proportional to t/ie distance of the
body from the centre ; all bodies revolving in any planes whatsoever
will describe ellipses, and complete their revolutions in equal times ;
and those which move in right lines, running backwards andforwards
alternately, will complete ttieir several periods of going and returning
in the same times.
For letting all things stand as in the foregoing Proposition, the force
SV, with which the body Q, revolving in any plane PQ,R is attracted to
wards the centre S, is as the distance SO. ; and therefore because SV and
SQ,, TV and CQ, are proportional, the force TV with which the body is
attracted towards the given point C in the plane of the orbit is as the dis
tance CQ,. Therefore the forces with which bodies found in the plane
PQ,R are attracted towaitis the point O, are in proportion to the distances
equal to the forces with which the same bodies are attract-ed every way to
wards the centre S ; and therefore the bodies will move in the same times,
and in the same figures, in any plane PQR about the point C. n* they
THE MATHEMATICAL PRINCIPLES [BOOK I.
would do in free spaces about the centre S ; and therefore (by Cor. 2, Prop.
X, ai d Cor. 2, Prop. XXXVIII.) they will in equal times either describe
ellipses m that plane about the centre C, or move to and fro in right lines
passing through the centre C in that plane; completing the same periods
of time in all cases. Q.E.D.
SCHOLIUM.
The ascent and descent of bodies in curve superficies has a near relation
to these motions we have been speaking of. Imagine curve lines to be de
scribed on any plane, and to revolve about any given axes passing through
the centre of force, and by that revolution to describe curve superficies ; and
that the bodies move in such sort that their centres may be always found
m those superficies. If those bodies reciprocate to and fro with an oblique
ascent and descent, their motions will be performed in planes passing through
tiie axis, and therefore in the curve lines, by whose revolution those curve
superficies were generated. In those cases, therefore, it will be sufficient to
consider thp motion in those curve lines.
PROPOSITION XLVIII. THEOREM XVI.
If a wheel stands npon the outside of a globe at right angles thereto, and
revolving about its own axis goes forward in a great circle, the length
of lite curvilinear path which any point, given in the perimeter of the
wheel, hath described, since, the time that it touched the globe (which
curvilinear path w~e may call the cycloid, or epicycloid), will be to double
the versed sine of half the arc which since that time has touched the
globe in passing over it, as the sn,m of the diameters of the globe and
the wheel to the semi-diameter of the globe.
PROPOSITION XLIX. THEOREM XVII.
ff a wheel stand upon the inside of a concave globe at right angles there
to, and revolving about its own axis go forward in one of the great
circles of the globe, the length of the curvilinear path which any point,
given in the perimeter of the wheel^ hath described since it toncJied the
globe, imll be to the double of the versed sine of half the arc which in
all that time has touched the globe in passing over it, as the difference
of the diameters of the globe and the wheel to the semi-diameter of the
globe.
Let ABL be the globe. C its centre, BPV the wheel insisting thereon,
E the centre of the wheel, B the point of contact, and P the given point
in the perimeter of the wheel. Imagine this wheel to proceed in the great
circle ABL from A through B towards L, and in its progress to revolve in
such a manner that the arcs AB, PB may be always equal one to the other,
:if;d the given point P in the peri meter of the wheel may describe in thf
SEC. X.I OF NATURAL PHILOSOPHY.
s
185
H
mean time the curvilinear path AP. Let AP be the whole curvilinear
path described since the wheel touched the globe in A, and the length cf
this path AP will be to twice the versed sine of the arc |PB as 20E to
CB. For let the right line CE (produced if need be) meet the wheel in V,
and join CP, BP, EP, VP ; produce CP, and let fall thereon the perpen
dicular VF. Let PH, VH, meeting in H, touch the circle in P and V,
and let PH cut YF in G, and to VP let fall the perpendiculars GI, HK.
From the centre C with any interval let there be described the circle wow,
cutting the right line CP in nt the perimeter of the wheel BP in o, and
the curvilinear path AP in m ; and from the centre V with the interval
Vo let there be described a circle cutting VP produced in q.
Because the wheel in its progress always revolves about the point of con
tact B. it is manifest that the right line BP is perpendicular to that curve line
AP which the point P of the wheel describes, and therefore that the right
line VP will touch this curve in the point P. Let the radius of the circle nmn
be gradually increased or diminished so that at last it become equal to the
distance CP ; and by reason of the similitude of the evanescent figure
Pnn-mq, and the figure PFGVI, the ultimate ratio of the evanescent lined ae
Pra, P//, Po, P<y,
that is, the ratio of the momentary mutations of the curve
AP, the right line CP, the circular arc BP, and the right line VP, will <
iSS THE MATHEMATICAL PRINCIPLES [BOOK 1.
the same as of the lines PV, PF, PG, PI, respectively. But since VF is
perpendicular to OF, and VH to CV, and therefore the angles HVG, VCF
equal: and the angle VHG (because the angles of the quadrilateral figure
HVEP are right in V and P) is equal to the angle CEP, the triangles
V HG, CEP will be similar ; and thence it will come to pass that as EP is
to CE so is HG to HV or HP, and so KI to KP, and by composition or
division as CB to CE so is PI to PK, and doubling the consequents asCB
to 2CE so PI to PV, and so is Pq to Pm. Therefore the decrement of the
line VP, that is, the increment of the line BY VP to the increment of the
curve line AP is in a given ratio of CB to 2CE, and therefore (by Cor.
Lena. IV) the lengths BY YP and AP, generated by those increments, arc
in the same ratio. But if BY be radius, YP is the cosine of the angle BYP
or -*BEP, and therefore BY YP is the versed sine of the same angle, and
therefore in this wheel, whose radius is ^BV, BY YP will be double the
versed sine of the arc ^BP. Therefore AP is to double the versed sine oi
the arc ^BP as 2CE to CB. Q.E.D.
The line AP in the former of these Propositions we shall name the cy
cloid without the globe, the other in the latter Proposition the cycloid within
the globe, for distinction sake.
COR. 1. Hence if there be described the entire cycloid ASL, and the
