tion) is either at rest, or moves uniformly in a right line. Let us first
suppose it at rest, and in s and p let there be placed two bodies, one im
movable in s, the other movable in p, similar and equal to the bodies S arid
P. Then let the right lines PR and pr touch the curves PQ, and pq ki P
and p, and produce CQ, and sq to R and r. And because the figures
CPRQ, sprq are similar, RQ, will be to rq as CP to sp, and therefore in a
given ratio. Hence if the force with which the body P is attracted to
wards the body S, and by consequence towards the intermediate point the
centre C, were to the force with which the body p is attracted towards the
Centre 5. in the same given ratio, these forces would in equal times attract
196 THE MATHEMATICAL PRINCIPLES |BoOK 1
the bodies from the tangents PR, pr to the arcs PQ, pq, through the in
tervals proportional to them RQ,, rq ; and therefore this last force (tending
to s) would make the body p revolve in the curve pqv, which would becomr
similar to the curve PQV, in which the first force obliges the body P i(
revolve ; and their revolutions would be completed in the same timeg
But because those forces are not to each other in the ratio of CP to sp, bu;
(by reason of the similarity and equality of the bodies S and s, P and /
and the equality of the distances SP, sp) mutually equal, the bodies ii
equal times will be equally drawn from the tangents; and therefore tLV
the body p may be attracted through the greater interval rq, there is re
quired a greater time, which will be in the subduplicate ratio of the inter
vals ; because, by Lemma X, the spaces described at the very beginning ol
the motion are in a duplicate ratio of the times. Suppose, then the velocity
of the body p to be to the velocity of the body P in a subduplicate ratio of
the distance sp to the distance CP, so that the arcs pq, PQ, which are in a
simple proportion to each other, may be described in times that are in n
subduplicate ratio of the distances ; and the bodies P, p, always attracted
by equal forces, will describe round the quiescent centres C and s similar
figures PQV, pqv, the latter of which pqv is similar and equal to the figure
ivhich the body P describes round the movable body S. Q.E.I)
CASE 2. Suppose now that the common centre of gravity, together with
the space in which the bodies are moved among themselves, proceeds uni
formly in a right line
; and (by Cor. 6, of the Laws of Motion) all the mo
tions in this space will be performed in the same manner as before ; and
therefore the bodies will describe mutually about each other the same fig
ures as before, which will be therefore similar and equal to the figure pqv.
Q.E.D.
COR. 1. Hence two bodies attracting each other with forces proportional
to their distance, describe (by Prop. X) both round their common centre ol
gravity, and round each other mutually concentrical ellipses ; and, vice
versa, if such figures are described, the forces are proportional to the dis
tances.
COR. 2. And two bodies, whose forces are reciprocally proportional to
the square of their distance, describe (by Prop. XI, XII, XIII), both round
their common centre of gravity, and round each other mutually, conic sec
tions having their focus in the centre about which the figures are described.
And, vice versa, if such figures are described, the centripetal forces are re
ciprocally proportional to the squares of the distance.
COR. 3. Any two bodies revolving round their common centre of gravity
describe areas proportional to the times, by radii drawn both to that centre
and to each other mutually
>EC. XL] OF NATURAL PHILOSOPHY. 197
PROPOSITION LIX. THEOREM XXII.
The periodic time of two bodies S and P revolving round their common
centre of gravity C,is to the periodic time of one of the bwlies 1? re
volving round the other S remaining unmoved, and describing a fig
ure similar and equal to those which the bodies describe about each
other mutuallyr
, in a subduplicate ratio of the other body S to the sii/rn
of the bodies S -f P.
For, by the demonstration of the last Proposition, the times in which
any similar arcs PQ and pq are described are in a subduplicate ratio of the
distances CP and SP, or sp, that is, in a subduplicate ratio of the ody S
to the sum of the bodies S + P. And by composition of ratios, the sums
of the times in which all the similar arcs PQ and pq are described, that is,
the whole times in which the whole similar figures are described are in the
same subduplicate ratio. Q.E.D.
PROPOSITION LX. THEOREM XXIII.
If two bodies S and P, attracting each other with forces reciprocally pro
portional to the squares of their distance, revolve about their common
centre of gravity ; I say, that the principal axis of the ellipsis which
either of the bodies, as P, describes by this motion about the other S,
will be to the principal axis of the ellipsis, which the same body P may
describe in the same periodical time about the other body S quiescent,
as the sum of the two bodies S + P to the first of two mean, propor
tionals between that sum and the other body S.
For if the ellipses described were equal to each other, their periodic times
by the last Theorem would be in a subduplicate ratio of the body S to the
sum of the bodies S 4- P. Let the periodic time in the latter ellipsis be
diminished in that ratio, and the periodic times will become equal ; but,
by Prop. XV, the principal axis of the ellipsis will be diminished in a ratio
sesquiplicate to the former ratio ; that is, in a ratio to which the ratio of
S to S 4- P is triplicate ; and therefore that axis will be to the principal
axis of the other ellipsis as the first of two mean proportionals between S
-f- P and S to S 4- P. And inversely the principal axis of the ellipsis de
scribed about the movable body will be to the principal axis of that described
round the immovable as S + P to the first of two mean proportionals be
tween S 4- P and S. Q.E.D.
PROPOSITION LXI. THEOREM XXIV.
If two bodies attracting each other with any kind of forces, and not
otherwise agitated or obstructed, are moved in any manner whatsoever,
those motions will be the same as if they did not at all attract each
other mutually, but were both attracted with the sameforces by a third
body placed in their common centre of gravity ; and the law of the
198 THE MATHEMATICAL PRINCIPLES [BOOK I
attracting Jones will be the sam# in respect of the distance of the.
bodies from, the common centre, as in respect of the distance between
the two bodies.
For those forces with which the bodies attract each other mutually, by
tending to the bodies, tend also to the common centre of gravity lying di
rectly between them ; and therefore are the same as if they proceeded from
an intermediate body. QJG.D.
And because there is given the ratio of the distance of either body from
that common centre to the distance between the two bodies, there is given,
of course, the ratio of any power of one distance to the same power of the
. ther distance ; and also the ratio of any quantity derived in any manner
from one of the distances compounded any how with given quantities, to
another quantity derived in like manner from the other distance, and as
many given quantities having that given ratio of the distances to the first
Therefore if the force with which one body is attracted by another be di
rectly or inversely as the distance of the bodies from each other, or as any
power of that distance ; or, lastly, as any quantity derived after any man
ner from that distance compounded with given q-uantities ; then will the
same force with which the same body is attracted to the common centre of
gravity be in like manner directly or inversely as the distance of the at
tracted body from the common centre, or as any power of that distance ;
or, lastly, as a quantity derived in like sort from that distance compounded
with analogous given quantities. That is, the law of attracting force will
be the same with respect to both distances. Q,.E.D.
PROPOSITION LXII. PROBLEM XXXVIII.
To determine the motions of two bodies which attract each other with
forces reciprocally proportional to the squares of the distance between
them, aflid are, let fallfrom given places.
The bodies, by the last Theorem, will be moved in the same manner as
if they were attracted by a third placed in the common centre of their
gravity ; and by the hypothesis that centre will be quiescent at the begin
ning of their motion, and therefore (by Cor. 4, of the Laws of Motion) will
be always quiescent. The motions of the bodies are therefore to be deter
mined (by Prob. XXV) in the same manner as if they were impelled by
forces tending to that centre: and then we shall have the motions of the
bodies attracting each other mutually. Q.E.I.
PROPOSITION LXIII. PROBLEM XXXIX.
To determine the motions of two bodies attracting each other with forces
reciprocally proportional to the squares of their distance, and going
offfrom given places in, given directions with given velocities.
The motions of the bodies at the beginning being given, there is given
SEC. XL] OF NATURAL PHILOSOPHY. 1%
also the uniform motion of the common centre of gravity, and the motion
of the space which moves along with this centre uniformly in a right line,
and also the very first, or beginning motions of the bodies in respect of this
space. Then (by Cor. 5, of the Laws, and the last Theorem) the subse
quent motions will be performed in the same manner in that space, as if
that space together with the common centre of gravity were at rest, and as
if the bodies did not attract each other, but were attracted by a third body
placed in that centre. The motion therefore in this movable space of each
body going off from a given place, in a given direction, with a given velo
city, and acted upon by a centripetal force tending to that centre, is to be
determined by Prob. IX and XXVI, and at the same time will be obtained
the motion of the other round the same centre. With this motion com
pound the uniform progressive motion of the entire system of the space and
the bodies revolving in it, and there will be obtained the absolute motion
of the bodies in immovable space. Q..E.I.
PROPOSITION LXIV. PROBLEM XL.
Supposingforces with which bodies mutually attract each other to in
crease in a simple ratio of their distances from the centres ; it is roquired
to find the motions of several bodies among themselves.
Suppose the first two bodies T and L
to have their common centre of gravity in
L). These, by Cor. 1, Theor. XXI, will
S
describe ellipses having their centres in D,
the magnitudes of which ellipses are
known by Prob. V.
J-
--
\- ? L
Let now a third body S attract the two
former T and L with the accelerative forces ST, SL, and let it be attract
ed again by them. The force ST (by Cor. 2, of the Laws of Motion) is
resolved into the forces SD, DT ; and the force SL into the forces SD and
DL. Now the forces DT, DL. which are as their sum TL, and therefore
as the accelerative forces with which the bodies T and L attract each other
mutually, added to the forces of the bodies T and L, the first to the first,
and the last to the last, compose forces proportional to the distances DT
and DL as before, but only greater than those former forces : and there
fore (by Cor. 1, Prop. X, and Cor. l,and 8, Prop. IV) they will cause those
bodies to describe ellipses as before, but with a swifter motion. The re
maining accelerative forces SD and DL, by the motive forces SD X Tand
SD X L, which are as the bodies attracting those bodies equally and in the
direction of the lines TI, LK parallel to DS, do not at all change their situ
ations with respect to one another, but cause them equally to approach to
the line IK ; which must be imagined drawn through the middle of the
body S, and perpendicular to the line DS. But that approach to the line
200 THE MATHEMATICAL PRINCIPLES [BoOK I.
IK will be hindered by causing the system of the bodies T and L on one
side, and the body S on the other, with proper velocities, to revolve round
the common centre of gravity C. With such a motion the body S, because
the sum of the motive forces SD X T and SD X L is proportional to the
distance OS, tends to the centre C, will describe an ellipsis round the same
centre C; and the point D, because the lines CS and CD are proportional,
will describe a like ellipsis over against it. But the bodies T and L, at
tracted by the motive forces SD X T and SD X L, the first by the first,
and the last by the last, equally and in the direction of the parallel lines TI
and LK, as was said before, will (by Cor. 5 and 6, of the Laws of Motion)
continue to describe their ellipses round the movable centre D, as before.
Q.E.I.
Let there be added a fourth body V, and, by the like reasoning, it will
be demonstrated that this body and the point C will describe ellipses about
the common centre of gravity B ; the motions of the bodies T, L, and S
round the centres D and C remaining the same as before ; but accelerated.
Arid by the same method one may add yet more bodies at pleasure. Q..E.I.
^This would be the case, though the bodies T and L attract each other
mutually with accelerative forces either greater or less than those with
which they attract the other bodies in proportion to their distance. Let
all the mutual accelerative attractions be to each other as the distances
multiplied into the attracting bodies ; and from what has gone before it
will easily be concluded that all the bodies will describe different ellipses
with equal periodical times about their common centre of gravity B, in an
immovable plane. Q.E.I.
PROPOSITION LXV. THEOREM XXV.
Bodies, whose forces decrease in a duplicate ratio of their distances from
their centres, may move among" themselves in ellipses ; and by radii
drawn to the foci may describe areas proportional to the times very
nearly.
In the last Proposition we demonstrated that case in which the motions
will be performed exactly in ellipses. The more distant the law of the
forces is from the law in that case, the more will the bodies disturb each
other s motions ; neither is it possible that bodies attracting each other
mutually according to the law supposed in this Proposition should move
exactly in ellipses, unless by keepirg a certain proportion of distances from
each other. However, in the following crises the orbits will not much dif
fer from ellipses.
CASE I. Imagine several lesser bodies to revolve about some very great
one at different distances from it, and suppose absolute forces tending to
rvery one of the bodies proportional to each. And because (by Cor. 4, ol
the I aws) the common centre of gravity of them all is either at rest, 01
iSEC. XL] OF NATURAL PHILOSOPHY. 20 i
moves uniformly forward in a right line, suppose the lesser bodies so small
that the groat body may be never at a sensible distance from that centre ;
and then the great body will, without any sensible error, be either at rest,
or move uniformly forward in a right line; and the lesser will revolve
about that great one in ellipses, and by radii drawn thereto will describe
areas proportional to the times ;
if we except the errors that may be intro
duced by the receding of the great body from the common centre of gravity,
or by the mutual actions of the lesser bodies upon each other. But the
lesser bodies may be so far diminished, as that this recess and the mutual
actions of the bodies on each other may become less than any assignable;
and therefore so as that the orbits may become ellipses, and the areas an
swer to the times, without any error that is not less than any assignable.
Q.E.O.
CASE 2. Let us imagine a system of lesser bodies revolving about a very
great one in the manner just described, or any other system of two bodies
revolving about each other to be moving uniformly forward in a right line, and
in the mean time to be impelled sideways by the force ofanother vastly greater
body situate at a great distance. And because the equal accelerative forces
with which the bodies are impelled in parallel directions do not change the
situation of the bodies with respect to each other, but only oblige the whole
system to change its place while the parts still retain their motions among
themselves, it is manifest that no change in those motions of the attracted
bodies can arise from their attractions towards the greater, unless by the
inequality of the accelerative attractions, or by the inclinations of the lines
towards each other, in whose directions the attractions are made. Suppose,
therefore, all the accelerative attractions made towards the great body
to be among themselves as the squares of the distances reciprocally ; and
then, by increasing the distance of the great body till the differences of fhe
right lines drawn from that to the others in respect of their length, and the
inclinations of those lines to each other, be less than any given, the mo
tions of the parts of the system will continue without errors that are not
less than any given. And because, by the small distance of those parts from
each other, the whole system is attracted as if it were but one body, it will
therefore be moved by this attraction as if it were one body ; that is, its
centre of gravity will describe about the great bod/ one of the conic sec
tions (that is, a parabola or hyperbola when the attraction is but languid
and an ellipsis when it is more vigorous) ; and by radii drawn thereto, it
will describe areas proportional to the times, without any errors but thos
which arise from the distances of the parts, which are by the supposition
exceedingly small, and may be diminished at pleasure. Q,.E.O.
By a like reasoning one may proceed to more compounded cases in infinitum.
COR 1. In the second Case, the nearer the very great body approaches to
^0^ THE MATHEMATICAL PRINCIPLES [CoOK I
the system of two or more revolving bodies, the greater will the pertur
bation be of the motions of the parts of the system among themselves; be
cause the inclinations of the lines drawn from that great body to those
parts become greater ; and the inequality of the proportion is also greater.
COR. 2. But the perturbation will be greatest of all, if we suppose the
uccelerative attractions of the parts of the system towards the greatest body
of all are not to each other reciprocally as the squares of the distances
from that great body ; especially if the inequality of this proportion be
greater than the inequality of the proportion of the distances from the
great body. For if the accelerative force, acting in parallel directions
and equally, causes no perturbation in the motions of the parts of the
system, it must of course, when it acts unequally, cause a perturbation some
where, which will be greater or less as the inequality is greater or less.
The excess of the greater impulses acting upon some bodies, and not acting
upon others, must necessarily change their situation among themselves. And
this perturbation, added to the perturbation arising from the inequality
and inclination of the lines, makes the whole perturbation greater.
COR. *. Hence if the parts of this system move in ellipses or circles
without any remarkable perturbation, it is manifest that, if they are at all
impelled by accelerative forces tending to any other bodies, the impulse is
very weak, or else is impressed very near equally and in parallel directions
upon all of them.
PROPOSITION LXVL THEOREM XXVI.
Tf three bodies whose forces decrease in a duplicate ratio of the distances
attract each other mutually ; and the accelerative attractions of any
two towards the third be between themselves reciprocally as the squares,
of the distances ; and the two least revolve about the greatest ; I say,
that the interior of the tivo revolving bodies will, by radii drawn to the
innermost and greatest, describe round thai body areas more propor
tional to the times, and a figure more approaching to that of an ellip
sis having its focus in the point of concourse of the radii, if that great
body be agitated by those attractions, than it would do if lhat great
body were not attracted at all by the lesser, but remained at rest ; or
than it would if that great body were very much more or very much
less attracted, <>r very much more or very much less agitated, by the
attractions.
This appears plainly enough from the demonstration of the second
Corollary of tl.e foregoing Proposition; but it may be made out after
this manner by a way of reasoning more distinct and more universally
convincing.
CASE 1. Let the lesser bodies P and S revolve in the same plane about
the greatest body T, the body P describing the interior orbit PAB, and S
SEC. XI.J OF NATURAL PHILOSOPHY. 203
the exterior orbit ESE. Let SK be the mean distance of the bodies P and
S ; and let the accelerative attraction of the body P towards S, at that
mean distance, be expressed by that line SK. Make SL to SK as the
E C
square of SK to the square of SP, and SL will be the accelerative attrac
tion of the body P towards S at any distance SP. Join PT, and draw
LM parallel to it meeting ST in M; and the attraction SL will be resolv
ed (by Cor. 2. of the Laws of Motion) into the attractions SM, LM. And
so the body P will be urged with a threefold accelerative force. One of
these forces tends towards T, and arises from the mutual attraction of the
bodies T and P. By this force alone the body P would describe round the
body T, by the radius PT, areas proportional to the times, and an
ellipsis whose focus is in the centre of the body T ; and this it would do
whether the body T remained unmoved, or whether it were agitated by that
attraction. This appears from Prop. XI, and Cor. 2 and 3 of Theor.
XXI. The other force is that of the attraction LM, which, because it
tends from P to T, will be superadded to and coincide with the former
force ; and cause the areas to be still proportional to the times, by Cor. 3,
Theor. XXI. But because it is not reciprocally proportional to the square
of the distance PT, it will compose, when added to the former, a force
varying from that proportion : which variation will be the greater by how
much the proportion of this force to the former is greater, cceteris paribus.
Therefore, since by Prop. XI, and by Cor. 2, Theor. XXI, the force with
which the ellipsis is described about the focus T ought to be directed to
that focus, and to be reciprocally proportional to the square of the distance
PT, that compounded force varying from that proportion will make the
orbit PAB vary from the figure of an ellipsis that has its focus in the point
I
1
; and so much the more by how much the variation from that proportion
is greater ; and by consequence by how much the proportion of the second
force LM to the first force is greater, cceteris paribus. But now the third
force SM, attracting the body P in a direction parallel to ST, composes with
the other forces a new force which is no longer directed from P to T : and which
varies so much more from this direction by how much the proportion of this
third force to the other forces is greater, cceterisparibus ; arid therefore causes
the body P to describe, by the radius TP, areas no longer proportional to the
times : and therefore makes the variation from that proportionality so much
greater by how much the proportion of this force to the others is greater.
But this third force will increase the variation of the orbit PAB from th*
THE MATHEMATICAL PRINCIPLES [BOOK 1
elliptical figure before-mentioned upon two accounts ;
first because that
force is not directed from P to T ; and, secondly, because it is not recipro
cally proportional to the square of the distance PT. These things being
premised, it is manifest that the areas are then most nearly proportional to
the times, when that third force is the least possible, the rest preserving
their former quantity ; and that the orbit PAB does then approach nearest
to the elliptical figure above-mentioned, when both the second and third,
but especially the third force, is the least possible; the first force remain
ing in its former quantity.
Let the accelerative attraction of the body T towards S be expressed by
the line SN ;
then if the accelerative attractions SM and SN were equal,
these, attracting the bodies T and P equally and in parallel directions
would not at all change their situation with respect to each other. The mo
tions of the bodies between themselves would be the same in that case as if
those attractions did not act at all, by Cor. 6, of the Laws of Motion. And,
by a like reasoning, if the attraction SN is less than the attraction SM, it
will take away out of the attraction SM the part SN, so that there will re
main only the part (of the attraction) MN to disturb the proportionality of
the areas and times, and the elliptical figure of the orbit. And in like
manner if the attraction SN be greater than the attraction SM, the pertur
bation of the orbit and proportion will be produced by the difference MN
alone. After this manner the attraction SN reduces always the attraction
SM to the attraction MN, the first and second attractions rema ning per
fectly unchanged ; and therefore the areas and times come then nearest to
proportionality, and the orbit PAB to the above-mentioned elliptical figure,
when the attraction MN is either none, or the least that is possible; that
is, when the accelerative attractions of the bodies P and T approach as near
as possible to equality ; that is, when the attraction SN is neither none at
all, nor less than the least of all the attractions SM, but is, as it were, a
mean between the greatest and least of all those attractions SM, that is,
not much greater nor much less than the attraction SK. Q.E.D.
CASE 2. Let now the lesser bodies P. S, revolve about a greater T in dif
