wards in the octants after the syzygies. And thence the greatest height of
the water may happen about the octants after the syzygies ; and the least
height about the octants after the quadratures ; excepting only so far as the
motion of ascent or descent impressed by these forces may by the vis insita
of the water continue a little longer, or be stopped a little sooner by impe
diments in its channel.
COR. 21. For the same reason that redundant matter in the equatorial
regions of a globe causes the nodes to go backwards, and therefore by the
increase of that matter that retrogradation is increased, by the diminution
is diminished, and by the removal quite ceases : it follows, that, if more than
214 THE MATHEMATICAL PRINCIPLES [BOOK I
that redundant matter be taken away, that is, if the globe be either more
depressed, or of a more rare consistence near the equator than near the
poles, there will arise a motion of the nodes in consequentia.
COR. 22. And thence from the motion of the nodes is known the consti
tution of the globe. That is, if the globe retains unalterably the same poles,
and the motion (of the nodes) be in. antecedetitia, there is a redundance oi
the matter near the equator; but if in conseqnentia, a deficiency. Sup
pose a uniform and exactly spherical globe to be first at rest in a free space :
then by some impulse made obliquely upon its superficies to be driven from
its place, and to receive a motion partly circular and partly right forward.
Because this globe is perfectly indifferent to all the axes that pass through
its centre, nor has a greater propensity to one axis or to one situation oi
the axis than to any other, it is manifest that by its own force it will never
change its axis, or the inclination of it. Let now this globe be impelled
obliquely by a new impulse in the same part of its superficies as before .
and since the effect of an impulse is not at all changed by its coming sooner
or later, it is manifest that these two impulses, successively impressed, will
produce the same motion as if they were impressed at the same time : that
is, the same motion as if the globe had been impelled by a simple force
compounded of them both (by Cor. 2, of the Laws), that is, a simple motion
about an axis of a given inclination. And the case is the same if the sec
ond impulse were made upon any other place of the equator of the first
motion ; and also if the first impulse were made upon any place in the
equator of the motion which would be generated by the second impulse
alone; and therefore, also, when both impulses are made in any places
whatsoever ; for these impulses will generate the same circular motion as
if they were impressed together, and at once, in the place of the intersec
tions of the equators of those motions, which would be generated by each
of them separately. Therefore, a homogeneous and perfect globe will not
retain several distinct motions, but will unite all those that are impressed
on it, and reduce them into one; revolving, as far as in it lies, always with
a simple and uniform motion about one single given axis, with an inclina
tion perpetually invariable. And the inclination of the axis, or the velocity
of the rotation, will not be changed by centripetal force. For if the globe
be supposed to be divided into two hemispheres, by any plane whatsoever
passing through its own centre, and the centre to which the force is direct
ed, that force will always urge each hemisphere equally ; and therefore will
not incline the globe any way as to its motion round its own axis. But
let there be added any where between the pole and the equator a heap oi
new matter like a mountain, and this, by its perpetual endeavour to recede
from the centre of its motion, will disturb the motion of the globe, and
cause its poles to wander about its superficies, describing circles about
themselves and their opposite points. Neither can this enormous evagatior
XL] OF NATURAL PHILOSOPHY. 2 In
of the poles be corrected, unless by placing that mountain ei . er in one ol
the poles; in which case, by Cor. 21, the nodes of the equator will go for
wards ;
or in the equatorial regions, in which case, by Cor. 20, the nodes
will go backwards: or, lastly, by adding on the other side of the axis anew
quantity of matter, by which the mountain may be balanced in its motion;
and then the nodes will either go forwards or backwards, as the mountain
and this newly added matter happen to be nearer to the pole or to the
equator.
PROPOSITION LXV1I. THEOREM XXVII.
The same laics of attraction being supposed, I say, that the exterior body
S does, by radii to the point O, the common centre of gravity
of the interior bodies P and T, describe round that centre areas more
proportional to the times, and an orbit more approaching to the form
of an ellipsis having its focus in that cen > .-. than, it can describe
round the innermost and greatest body T by ra Hi drawn to that
body.
For the attractions of the body S towards T and
P compose its absolute attraction, which is more
directed towards O, the common centre of gravity
S(i
of the bodies T and P, than it is to the . reatest
body T ; and which is more in a reciprocal propor
tion to the square of the distance SO, than it is to the square of the distance
ST : as will easily appear by a little consideration.
PROPOSITION LXVIII. THEOREM XXVIII.
The same laws of attraction supposed, I say, that the exterior body S
will, by radii drawn to O, the common centre of gravity of the interior
bodies P and T, describe round that centre areas more propor
tional to the times, and an orbit more approaching to the form of an
ellipsis having its focus in that centre, if the innermost and greatest
body be agitated by these attractions as well as the rest, than it would
do if that body were either at rest as not attracted, or were much tnore
or much less attracted, or much more or much less agitated.
This may be demonstrated after the same manner as Prop. LXVI, but
by a more prolix reasoning, which I therefore pass over. It will be suf
ficient to consider it after this manner. From the demonstration of the
last Proposition it is plain, that the centre, towards which the body S is
urged by the two forces conjunctly, is very near to the common centre of
gravity of those two other bodies. If this centre were to coincide with that
common centre, and moreover the common centre of gravity of all the three
bodies were at rest, the body S on one side, and the common centre of
gravity of the other two bodies on the other side, would describe true ellip*
216 THE MATHEMATICAL PRINCIPLES - [BOOK 1
ses about that quiescent common centre. This appears from Cor. 2, Pro])
LVIII, compared with what was demonstrated in Prop. LX1V, and LXY
Now this accurate elliptical motion will be disturbed a little by the dis
tance of the centre of the two bodies from the centre towards which tht
third body S is attracted. Let there be added, moreover, a motion to the
Bommon centre of the three, and the perturbation will be increased yet
more. Therefore the perturbation is least when the
common centre of the three bodies is at rest; that
I is, when the innermost and greatest body T is at
tracted according to the same law as the rest are ;
and is always greatest when the common centre of
the three, by the diminution of the motion of the body T, begins to be
moved, and is more and more agitated.
COR. And hence if more lesser bodies revolve about the great one, it
may easily be inferred that the orbits described will approach nearer to
ellipses ; and the descriptions of areas will be more nearly equable, if all
the bodies mutually attract and agitate each other with accelerative forces
that are as their absolute forces directly, and the squares of the distances
inversely : and if the focus of each orbit be placed in the common centre
of gravity of all the interior bodies (that is. if the focus of the first and in
nermost orbit be placed in the centre of gravity of the greatest and inner
most body : the focus of the second orbit in the common centre of gravity
of the two innermost bodies; the focus of the third orbit in the common
centre of gravity of the three innermost ; and so on), than if the innermost
body were at rest, and was made the common focus of all the orbits.
PROPOSITION LXIX. THEOREM XXIX.
fn a system of several bodies A, B, C, D, $*c., if any one of those bodies,
as A, attract all the rest, B, C, D, $*c.,with accelerative forces that are
reciprocally as the squares of the distances from the attracting body ;
and another body, as B, attracts also the rest. A, C, D, $-c., with forces
that are reciprocally as the squares of the distances from the attract
ing body ; the absolute forces of the attracting bodies A and B will
be to each other as those very bodies A and B to which those forces
belong.
For the accelerative attractions of all the bodies B, C, D, towards A,
are by the supposition equal to each other at equal distances ; and in like
manner the accelerative attractions of all the bodies towards B are also
equal to each other at equal distances. But the absolute attractive force
of the body A is to the absolute attractive force of the body B as the aceelerative
attraction of all the bodies towards A to the accelerative attrac
tion of all the bodies towards B at equal distances ; and so is also the ac
celerative attraction of the body B to*vards A to the accelerative attraction
SEC. XI] OF NATURAL PHILOSOPHY. 21 T
of the body A towards B. But the accelerative attraction of the body B
towards A is to the accelerative attraction of the body A towards B as the
mass of the body A to the mass of the body B ; because the motive forces
which (by the 2d, 7th, and 8th Definition) are as the accelerative forces
and the bodies attracted conjunctly are here equal to one another by the
third Law. Therefore the absolute attractive force of the body A is to the
absolute attractive force of the body B aa the mass of the body A to the
mass of the body B. Q.E.D.
COR. 1. Therefore if each of the bodies of the system A, B, C, D, &c.
does singly attract all the rest with accelerative forces that are reciprocally
as the squares of the distances from the attracting body, the absolute forces
of all those bodies will be to each other as the bodies themselves.
COR. 2. By a like reasoning, if each of the bodies of the system A, B,
C, D, &c., do singly attract all the rest with accelerative forces, which are
either reciprocally or directly in the ratio of any power whatever of the
distances from the attracting body : or which are defined by the distances
from each of the attracting bodies according to any common law : it is plain
that the absolute forces of those bodies are as the bodies themselves.
COR. 3. In a system of bodies whose forces decrease in the duplicate ra
tio of the distances, if the lesser revolve about one very great one in ellip
ses, having their common focus in the centre of that great body, and of a
figure exceedingly accurate ; and moreover by radii drawn to that great
ody describe areas proportional to the times exactly the absolute forces
)i those bodies to each other will be either accurately or very nearly in the
ratio of the bodies. And s > on the contrary. This appears from Cor. of
Prop. XLVII1, compared with the first Corollary of this Prop.
SCHOLIUM.
These Propositions naturally lead us to the analogy there is between
centripetal forces, and the central bodies to which those forces used to be
directed ; for it is reasonable to suppose that forces which are directed to
bodies should depend upon the nature and quantity of those bodies, as we
see they do in magnetical experiments. And when such cases occur, we
are to compute the attractions of the bodies by assigning to each of their
particles its proper force, and then collecting the sum of them all. I here
ue*e the word attraction in general for any endeavour, of what kind soever,
made by bodies to approach to each other; whether that endeavour arise
from the action of the bodies themselves, as tending mutually to or agita
ting each other by spirits emitted; or whether it arises from the action
of the aether or of the air, or of any medium whatsoever* whether corporeal
or incorporeal, any how impelling bodies placed therein towards each other.
In the same general sense I use the word impulse, not defining in this trea
tise the species or physical qualities of forces, but investigating the quantities
THE MATHEMATICAL PRINCIPLES [BOOK ).
and mathematical proportions of them ; as I observed before ir (lie Defi
nitions. In mathematics we are to investigate the quantities of forces
with their proportions consequent upon any conditions supposed ; then,
when we enter upon physics, we compare those proportions with the phe
nomena of Nature, that we may know what conditions of those forces an
swer to the several kinds of attractive bodies. And this preparation being
made, we argue more safely concerning the physical species, causes, and
proportions of the forces. Let us see, then, with what forces spherical
bodies consisting of particles endued with attractive powers in the manner
above spoken of must act mutually upon one another : and what kind of
motions will follow from thence.
SECTION XII.
Of the attractive forces of sphcerical bodies.
PROPOSITION LXX. THEOREM XXX.
If to every point of a spherical surface there tend equal centripetal forces
decreasing in, the duplicate ratio of the distances from those points ;
I say, that a corpuscle placed within that superficies will not be attract
ed by those forces any way.
Let HIKL, be that sphaerical superficies, and P a
corpuscle placed within. Through P let there be
drawn to this superficies to two lines HK, IL, intercepting
very small arcs HI, KL ; and because (by
Cor. 3, Lem. VII) the triangles HPI,LPK are alike,
those arcs will be proportional to the distances HP
LP ; and any particles at HI and KL of the spheri
cal superficies, terminated by right lines passing through P, will be in the
duplicate ratio of those distances. Therefore the forces of these particles
exerted upon the body P are equal between themselves. For the forces are
as the particles directly, and the squares of the distances inversely. And
these two ratios compose the ratio of equality. The attractions therefore,
being made equally towards contrary parts, destroy each other. And by a
like reasoning all the attractions through the whole spherical superficies
are destroyed by contrary attractions. Therefore the body P will not be
any way impelled by those attractions. Q.E.D.
PROPOSITION LXXI. THEOREM XXXI.
The same things supposed as above, I say, that a corpu vie placed with
out the sph(ericl superficies is attracted towards the centre of tht
sphere wiih a force reciprocally proportional to the square of its dis
tance from that centre.
Let AHKB, ahkb, be two equal sphaerical superficies described about
SEC. XII.J OF NATURAL PHILOSOPHY.
the centre S, s ; their diameters AB, ab ; and let P and p be two corpus
cles situate without the gpheres in those diameters produced. Let there
be drawn from the corpuscles the lines PHK, PIL, phk, pil, cutting off
from the great circles AHB, ahb, the equal arcs HK, hk, IL;
il ; and to
those lines let fall the perpendiculars SD, sd, SE, SP, 1R, ir ; of which let
SD, sd, cut PL, pi, in F and f. Let fall also to the diameters the perpen
diculars IQ, iq. Let now the angles DPE, dpe, vanish; and because DS
and ds, ES and es are equal, the lines PE, PP, and pe, pf, and the lineolso
I )F, df may be taken for equal ; because their last ratio, when the angles
DPE, dpe vanish together, is the ratio of equality. These things then
supposed, it will be, as PI to PF so is RI to DF, and as pf to pi so is df or
DF to ri ; and, ex cequo, as PI X pf to PF X pi so is RI to ri, that is
(by Cor. 3, Lem VII), so is the arc IH to the arc ih. Again, PI is to PS
as IQ. to SE, and ps to pi as se or SE to iq ; and, ex ceqno, PI X ps to
PS X pi as IQ. to iq. And compounding the ratios PI 2 X pf X ps is to
pi
2 X PF X PS, as IH X IQ to ih X iq ; that is, as the circular super
ficies which is described by the arc IH, as the semi-circle AKB revolves
about the diameter AB, is to the circular superficies described by the arc ih
as the semi-circle akb revolves about the diameter ab. And the forces
with which these superficies attract the corpuscles P and p in the direction
of lines tending to those superficies are by the hypothesis as the superficies
themselves directly, and the squares of the distances of the superficies from
those corpuscles inversely; that is, as pf X ps to PF XPS. And these
forces again are to the oblique parts of them which (by the resolution of
forces as in Cor. 2, of the Laws) tend to the centres in the directions of the
lines PS, JDS-, as PI to PQ, and pi to pq ; that is (because of the like trian
gles PIQ and PSF, piq and psf\ as PS to PF and ps to pf. Thence ex
cequO) the attraction of the corpuscle P towards S is to the attraction of
PF XpfXps. pf X PF X PS .
the corpusclejo towards 5 as ~ = is to , that is,
as ps
2 to PS2
. And, by a like reasoning, the forces with which the su
perficies described by the revolution of the arcs KL, kl attract those cor
puscles, will be as jDS
2 to PS2
. And in the same ratio will be the foroes
of all the circular superficies into which each of the sphaerical superficies
may be divided by taking sd always equal to SD, and se equal to SE. And
therefore, by composition, the forces of the entire spherical superficies ex
erted upon those corpuscles will be in the same ratio. Q.E.D
220 THE MATHEMATICAL PRINCIPLES [BOOK 1
PROPOSITION LXXIL THEOREM XXXII.
If to the several points of a sphere there tend equal centripetal forces de
creasing in a duplicate ratio of the distances from those points ; and
there be given both the density of the sphere and the ratio of the di
ameter of the sphere to the distance of the corpuscle from its centre ;
I say, that the force with which the corpuscle is attracted is propor
tional to the semi-diameter of the sphere.
For conceive two corpuscles to be severally attracted by two spheres, one
by one, the other by the other, and their distances from the centres of the
spheres to be proportional to the diameters of the spheres respectively , and
the spheres to be resolved into like particles, disposed in a like situation
to the corpuscles. Then the attractions of one corpuscle towards the sev
eral particles of one sphere will be to the attractions of the other towards
as many analogous particles of the other sphere in a ratio compounded of
the ratio of the particles directly, and the duplicate ratio of the distances
inversely. But the particles are as the spheres, that is, in a triplicate ra
tio of the diameters, and the distances are as the diameters ; and the first
ratio directly with the last ratio taken twice inversely, becomes the ratio
of diameter to diameter. Q.E.D.
COR. 1. Hence if corpuscles revolve in circles about spheres composed
of matter equally attracting, and the distances from the centres of the
spheres be proportional to their diameters, the periodic times will be equal.
COR. 2. And, vice versa, if the periodic times are equal, the distances
will be proportional to the diameters. These two Corollaries appear from
Cor. 3, Prop. IV.
COR. 3. If to the several points of an^ two solids whatever, of like figare
and equal density, there tend equal centripetal forces decreasing in a
duplicate ratio of the distances from those points, the forces, with which
corpuscles placed in a like situation to those two solids will be attracted
by them, will be to each other as the diameters of the solids.
PROPOSITION LXXIII. THEOREM XXXIII.
If to the several points of a given sphere there tend equal centripetal forces
decreasing in a duplicate ratio of the distances from the points ; 1
say, that a corpuscle placed within the sphere is attracted by a force
proportional to its distancefrom the centre.
In the sphere ABCD, described about the centre S,
let there be placed the corpuscle P ; and about the
same centre S, with the interval SP? conceive de-
|
B scribed an interior sphere PEQP. It is plain (by
Prop. LXX) that the concentric sphaerical superficies,
of which the difference AEBF of the spheres is com
posed, have no effect at all upon the body P, their at
SEC. XIL] OF NATURAL PHILOSOPHY. 22\
tractions being destroyed by contrary attractions. There remains, there
fore; only the attraction of the interior sphere PEQ,F. And (by Prop.
LXXII) this is as the distance PS. Q.E.D.
SCHOLIUM.
By the superficies of which I here imagine the solids composed, I do not
mean superficies purely mathematical, but orbs so extremely thin, that
their thickness is as nothing; that is, the evanescent orbs of which the sphere
will at last consist when the number of the orbs is increased, and their
thickness diminished without end. In like manner, by the points of which
lines, surfaces, and solids are said to be composed, are to be understood
equal particles, whose magnitude is perfectly inconsiderable.
PROPOSITION LXXIV. THEOREM XXXIV.
The same things supposed, I say, that a corpuscle situate without the
sphere is attracted with a force reciprocally proportional to the square
of its distance from the centre.
For suppose the sphere to be divided into innumerable concentric sphe
rical superficies, and the attractions of the corpuscle arising from the sev
eral superficies will be reciprocally proportional to the square of the dis
tance of the corpuscle from the centre of the sphere (by Prop. LXXI).
And, by composition, the sum of those attractions, that is, the attraction
of the corpuscle towards the entire sphere, will be in the same ratio. Q.E.D.
COR. 1. Hence the attractions of homogeneous spheres at equal distances
from the centres will be as the spheres themselves. For (by Prop. LXXII)
if the distances be proportional to the diameters of the spheres, the forces
will be as the diameters. Let the greater distance be diminished in that
ratio ; and the distances now being equal, the attraction will be increased
in the duplicate of that ratio ; and therefore will be to the other attraction
in the triplicate of that ratio
;
that is, in the ratio of the spheres.
COR. 2. At any distances whatever the attractions are as the spheres
applied to the squares of the distances.
COR. 3. If a corpuscle placed without an homogeneous sphere is attract
ed by a force reciprocally proportional to the square of its distance from
the centre, and the sphere consists of attractive particles, the force of every
particle will decrease in a duplicate ratio of the distance from each particle.
PROPOSITION LXXV. THEOREM XXXV.
If to the several points of a given sphere there tend equal centripetal forces
decreasing in a duplicate ratio of the distances from the points ; Isay,
that another similar sphere will be attracted by it with a force recip
rocally proportional to the square of the distance of the centres.
For the attraction of every particle is reciprocally as the square of its
222 THE MATHEMATICAL PRINCIPLES | BOOK L
distance from the centre of the attracting sphere (by Prop. LXXIV). and
is therefore the same as if that whole attracting force issued from one sin
gle corpuscle placed in the centre of this sphere. But this attraction is as
great as on the other hand the attraction of the same corpuscle would be,
if that were itself attracted by the several particles of the attracted sphere
with the same force with which they are attracted by it. But that attrac
tion of the corpuscle would be (by Prop. LXXIV) reciprocally propor
tional to the square of its distance from the centre of the sphere : therefore
the attraction of the sphere, equal thereto, is also in the same ratio. Q,.E.D.
COR. 1. The attractions of spheres towards other homogeneous spheres
are as the attracting spheres applied to the squares of the distances of their
centres from the centres of those which they attract.
COR. 2. The case is the same when the attracted sphere does also at
tract. For the several points of the one attract the several points of the
other with the same force with which they themselves are attracted by the
others again; and therefore since in all attractions (by Law III) the at
tracted and attracting point are both equally acted on, the force will be
doubled by their mutual attractions, the proportions remaining.
COR. 3. Those several truths demonstrated above concerning the motion
of bodies about the focus of the conic sections will take place when an
attracting sphere is placed in the focus, and the bodies move without the
sphere.
COR. 4. Those things which were demonstrated before of the motion of
bodies about the centre of the conic sections take place when the motions
are performed within the sphere.
PROPOSITION LXXVI. THEOREM XXXVI.
ff spheres be however dissimilar (as to density of matter and attractive,
force] in the same ratio onwardfrom the centre to the circumference ;
but every where similar, at every given distance from the centre, on all
sides round about ; and the attractive force of every point decreases
in the duplicate ratio of the distance of the body attracted ; I say,
that the whole force with which one of these spheres attracts the oilier
will be reciprocally proportional to the square of the distance of the
centres.
