As if the centripetal forces of the
particles of the sphere be reciprocally
;is the distances of the corpuscle at
tracted by them ; the force with which
the corpuscle situate in I is attracted
by the entire sphere will be to the
force with which it is attracted in P
in a ratio compounded of the subdu
plicate ratio of the distance SI to the distance SP, and the subduplicate
ratio of the centripetal force in the place I arising from any particle in the
centre to the centripetal force in the place P arising from the same particle in
the centre ; that is, in the subduplicate ratio of the distances SI, SP to each
other reciprocally. These two subduplicate ratios compose the ratio of
equality, and therefore the attractions in I and P produced by the whole
sphere are equal. By the like calculation, if the forces of the particles of
the sphere are reciprocally in a duplicate ratio of the distances, it will be
found that the attraction in I is to the attraction in P as the distance SP
to the semi -diameter SA of the sphere. If those forces are reciprocally in
a triplicate ratio of the distances, the attractions in I and P will be to each
other as SP 2 to SA3
;
if in a quadruplicate ratio, as SP 3 to SA 3
. There
fore since the attraction in P was found in this last case to be reciprocally
as PS 3 X PI, the attraction in I will be reciprocally as SA 3 X PI, that is,
because SA 3 is given reciprocally as PI. And the progression is the same
in injinitnm. The demonstration of this Theorem is as follows :
The things remaining as above constructed, and a corpuscle being in anj
332 THE MATHEMATICAL PRINCIPLES [BOOK I.
place P. the ordinate DN was found to be as
T)F 2
\" PS
00 ^~
- Therefore if
I Cj X V
IE be drawn, that ordinate for any other place of the corpuscle, as I, will
DE2 X IS
become (mutatis mutandis] as
~T~p~rry~- Suppose the centripetalsorces
flowing from any point of the sphere, as E, to be to each other at the dis
tances IE and PE as PE 1 to IE11
(where the number u denotes the index
DE 2 X PS
of the powers of PE and IE), and those ordinates will become as ^p--57^7,
2 \x IS
and
~"---
TT7, whose ratio to each other is as PS X IE X IEn to IS X
IE X IE"
PE X PEn
. Because SI, SE, SP are in continued proportion, the tri
angles SPE, SEI are alike ; and thence IE is to PE as IS to SE or SA.
For the ratio of IE to PE write the ratio of IS to SA ; and the ratio of
the ordinates becomes that of PS X IE" to SA X PEn
. But the ratio of
PS to SA is snbduplicate of that of the distances PS, SI ; and the ratio of
IE" to PE 1
(because IE is to PE as IS to SA) is subduplicate of that of
the forces at the distances PS, IS. Therefore the ordinates, and conse
quently the areas whioifi the ordinates describe, and the attractions propor
tional to them, are in a ratio compounded of those subduplicate ratios.
Q.E.D.
PROPOSITION LXXXIII. PROBLEM XLII.
To find the force with which a corpuscle placed in the centre of a sphere
is attracted towards any segment of that sphere whatsoever.
Let P be a body in the centre of that sphere and
RBSD a segment thereof contained under the plane
RDS, and thesphrcrical superficies RBS. Let DB be cut
in F by a sphaerical superficies EFG described from the
centre P, and let the segment be divided into the parts
_B BREFGS, FEDG. Let us suppose that segment to
be not a purely mathematical but a physical superficies,
having some, but a perfectly inconsiderable thickness.
* Let that thickness be called O, and (by what Archi
medes has demonstrated) that superficies will be as
PF X DF X O. Let us suppose besides the attrac
tive forces of the particles of the sphere to be reciprocally as that power of
distances, of which n is index ; and the force with which the superficies
DE2 X O
EFG attracts the body P will be (by Prop. LXXIX) as -- that,
2DF X O
is, as ---? -,-
DF 2 X O
~"~ppn
*
ppn
the perpendicular FN drawn into
SEC. XJ11.I OF NATURAL PHILOSOPHY. 233
O be proportional to this quantity ; and the curvilinear area BDI, which
the ordinate FN, drawn through the length DB with a continued motion
will describe, will be as the whole force with which the whole segment
RBSD attracts the body P. Q.E.I.
PROPOSITION LXXXIV. PROBLEM XLIII.
To find the force with which a corpuscle, placed without the centre of a
sphere iti the axis of any segment, is attracted by that segment.
Let the body P placed in. the axis ADB of
the segment KBK be attracted by that seg
ment. About the centre P, with the interval
PE, let the spherical superficies EFK be described;
and let it divide the segment into
two parts EBKFE and EFKDE. Find the
force of the first of those parts by Prop.
LXXXI, and the force of the latter part by
Prop. LXXXIII, and the sum of the forces will be the force of the whole
segment EBKDE. Q.E.I.
SCHOLIUM.
The attractions of sphaerical bodies being now explained, it comes next
in order to treat of the laws of attraction in other bodies consisting in like
manner of attractive particles ; but to treat of them particularly is not neces
sary to my design. It will be sufficient to subjoin some general proposi
tions relating to the forces of such bodies, and the motions thence arising,
because the knowledge of these will be of some little use in philosophical
inquiries.
SECTION XIII.
Of the attractive forces of bodies which are not of a sphcerical figure.
PROPOSITION LXXXV. THEOREM XLIL
If a body be attracted by another, and its attraction be vastly stronger
when it is contiguous to the attracting body than when they are sepa
rated from one another by a very small interval ; the forces of the
particles of the attracting body decrease, in the recess of the body at
tracted, in more than a duplicate ratio of the distance of the particles.
For if the forces decrease in a duplicate ratio of the distances from the
particles, the attraction towards a sphaerical body being (by Prop. LXXIV)
reciprocally as the square of the distance of the attracted body from the
centre of the sphere, will not be sensibly increased by the contact, and it
234 THE MATHEMATICAL PRINCIPLES [BOOK 1
\vill be still less increased by it, if the attraction, in the recess of the body
attracted, decreases in a still less proportion. The proposition, therefore,
is evident concerning attractive spheres. And the case is the same of con
cave sphaerical orbs attracting external bodies. And much more does it
appear in orbs that attract bodies placed within them, because there the
attractions diffused through the cavities of those orbs are (by Prop. LXX)
destroyed by contrary attractions, and therefore have no effect even in the
place of contact. Now if from these spheres and sphoerical orbs we take
away any parts remote from the place of contact, and add new parts any
where at pleasore, we may change the figures of the attractive bodies at
pleasure ; but the parts added or taken away, being remote from the place
of contact, will cause no remarkable excess of the attraction arising from
the contact of the two bodies. 1 herefore the proposition holds good in
bodies of all figures. Q.E.I).
PROPOSITION LXXXV1. THEOREM XLIII.
If the forces of the particles of which an attractive body is composed de
crease, in. the recess of the attractive body, in a triplicate or more than
a triplicate ratio of the distancefrom the particles, the attraction will
be vastly stronger in the point of contact than when the attracting and
attracted bodies are separated from each other, though by never so
small an interval.
For that the attraction is infinitely increased when the attracted corpus
cle comes to touch an attracting sphere of this kind, appears, by the solu
tion of Problem XLI, exhibited in e second and third Examples. The
same will also appear (by comparing those Examples and Theorem XLI
together) of attractions of bodies made towards concavo-convex orbs, whether
the attracted bodies be placed without the orbs, or in the cavities within
them. And by aiding to or taking from those spheres and orbs any at
tractive matter any where without the place of contact, so that the attrac
tive bodies may receive any assigned figure, the Proposition will hold good
of all bodies universally. Q.E.D.
PROPOSITION LXXXVII. THEOREM XI. IV.
If two bodies similar to each other, and consisting of matter equally at
tractive attract separately two corpuscles proportional to those bodies,
and in a like situation to them, the accelerative attractions of the cor
puscles towards the entire bodies will be as the acccleratire attractions
of the corpuscles towards particles of the bodies proportional to the
wholes, and alike situated in them.
For if the bodies are divided into particles proportional to the wholes,
and alike situated in them, it will be, as the attraction towards any parti
cle of one of the bodies to the attraction towards the correspondent particle
SEC. A III.] OF NATURAL PHILOSOPHY. 235
in the other body, so are the attractions towards the several particles of the
iirst body, to the attractions towards the several correspondent particles of
the other body ; and, by composition, so is the attraction towards the first
whole body to the attraction towards the second whole body. Q,.E.U.
COR. 1 . Therefore if, as the distances of the corpuscles attracted increase,
the attractive forces of the particles decrease in the ratio of any power
of the distances, the accelerative attractions towards the whole bodies will
be as the bodies directly, and those powers of the distances inversely. A*
if the forces of the particles decrease in a duplicate ratio of the distances
from the corpuscles attracted, and the bodies are as A 3 and B 3
, and there
fore both the cubic sides of the bodies, and the distance of the attracted
corpuscles from the bodies, are as A and B ; the accelerative attractions
A 3 B 3
towards the bodies will be as and , that is, as A and B the cubic
sjides of those bodies. If the forces of the particles decrease in a triplicate
ratio of the distances from the attracted corpuscles, the accelerative attrac-
A3 B 3
tions towards the whole bodies will be as and 5--, that is, equal. If the
A. tj
forces decrease in a quadruplicate ratio, the attractions towards the bodies
A 3 B 3
will be as- an^ 04 *^at is, reciprocally as the cubic sides A and B.
And so in other cases.
COR. 2. Hence, on the other hand, from the forces with which like bodies
attract corpuscles similarly situated, may be collected the ratio of the de
crease of the attractive forces of the particles as the attracted corpuscle
recedes from them ;
if so be that decrease is directly or inversely in any
ratio of the distances.
PROPOSITION LXXXVIII. THEOREM XLV.
If the attractive forces of the equal particles of any body be as the dis
tance of the places from the particles, the force of the whole body will
tend to its centre of gravity ; and will be the same with the force of
a globe, consisting of similar and equal matter, and having its centre
in the centre of gravity.
Let the particles A, B, of the body RSTV at
tract any corpuscle Z with forces which, suppos-|
ing the particles to be equal between themselves,
are as the distances AZ, BZ ; but, if they are
supposed unequal, are as those particles and
their distances AZ, BZ, conjunctly, or (if I may
go speak) as those particles drawn into their dis
tances AZ, BZ respectively. And let those forces be expressed by the
236 THE MATHEMATICAL PRINCIPLES [BOOK 1.
contents er A X AZ, and B X BZ. Join AB, and let it be cut in G,
so that AG may be to BG as the particle B to the particle A : and G
will be the common centre of gravity of the particles A and B. The force
A X AZ will (by Cor. 2, of the Laws) be resolved into the forces A X GZ
and A X AG ; and the force B X BZ into the forces B X GZ and B X
BG. Now the forces A X AG and B X BG, because A is proportional to
B, and BG to AG, are equal, and therefore having contrary directions de
stroy one another. There remain then the forces A X GZ and B X GZ.
These tend from Z towards the centre G, and compose the force A + B
X GZ ; that is, the same force as if the attractive particles A and B were
placed in their common centre of gravity G, composing there a little globe.
By the same reasoning, if there be added a third particle G, and the
force of it be compounded with the force A -f B X GZ tending to the cen
tre G, the force thence arising will tend to the common centre of gravity
of that globe in G and of the particle C ; that is, to the common centre oi
gravity of the three particles A, B, C ; and will be the same as if that
globe and the particle C were placed in that common centre composing a
greater globe there ; and so we may go on in injinitum. Therefore
the whole force of all the particles of any body whatever RSTV is the
same as if that body, without removing its centre of gravity, were to put
on the form of a globe. Q,.E.D.
COR. Hence the motion of the attracted body Z will be the same as if
the attracting body RSTV were sphaerical ; and therefore if that attract
ing body be either at rest, or proceed uniformly in a right line, the body
attracted will move in an ellipsis having its centre in the centre of gravity
of the attracting body.
PROPOSITION LXXXIX. THEOREM XLVI.
If there be several bodies consisting of equal particles whose jorces are
as the distances of the places from each, the force compounded of all
the forces by which any corpuscle is attracted will tend to the common
centre of gravity of the attracting bodies ; and will be the same as if
those attracting bodies, preserving their common centre of gravity,
should unite there, and be formed into a globe.
This is demonstrated after the same manner as the foregoing Proposi
tion.
COR. Therefore the motion of the attracted body will be the same as if
the attracting bodies, preserving their common centre of gravity, should
unite there, and be formed into a globe. And, therefore, if the common
centre of gravity of the attracting bodies be either at rest, or proceed uni
formly in a right line, the attracted body will move in an ellipsis having
Us centre in the common centre of gravity of the attracting bodies.
SEC. XlII.j OF NATURAL PHILOSOPHY. 237
PROPOSITION XC. PROBLEM XLIV.
If to the several points of any circle there tend equal centripeta forces,
increasing or decreasing in any ratio of the distances ; it is required
to Jind the force with which a corpuscle is attracted, that is, situate
any where in a right line which stands at right angles to the plant
of the circle at its centre.
Suppose a circle to be described about the cen
tre A with any interval AD in a plane to which ;
the right line AP is perpendicular ; and let it be
required to find the force with which a corpuscle
P is attracted towards the same. From any point
E of the circle, to the attracted corpuscle P, let
there be drawn the right line PE. In the right
line PA take PF equal to PE, and make a perpendicular
FK, erected at F, to be as the force
with which the point E attracts the corpuscle P.
And let the curve line IKL be the locus of the point K. Let that cu/, fe
meet the plane of the circle in L. In PA take PH equal to PD, and p/^ct
the perpendicular HI meeting that curve in I
; and the attraction of the
corpuscle P towards the circle will be as the area AHIL drawn into the
altitude AP Q.E.I.
For let there be taken in AE a very small line Ee. Join Pe, and in PE,
PA take PC, Pf equal to Pe. And because the force, with which any
point E of the annulus described about the centre A with the interval AS
in the aforesaid plane attracts to itself the body P, is supposed to be as
FK ; and, therefore, the force with which that point attracts the body P
AP X FK
towards A is as -
^p ; and the force with which the whole annulus
AP X FK
attracts tne body P towards A is as the annulus and p^ conjunctly
; and that annulus also is as the rectangle under the radius AE aad the
breadth Ee, and this rectangle (because PE and AE, Ee and CE are pro
portional) is equal to the rectangle PE X CE or PE X F/; the force
*-ith which that annulus attracts the body P towards A will be as PE X
AP X FK
Ff and pp~~~ conjunctly ; that is, as the content under F/ X FK X
AP, or as the area FKkf drawn into AP. And therefore the sum of the
forces with which all the annuli, in the circle described about the centre A
with the interval AD, attract the body P towards A, is as the whole area
AHIKL drawn into AP. Q.E.D.
COR. 1. Hence if the forces of the points decrease in the duplicate ratio
238 THE MATHEMATICAL PRINCIPLES [BOOK I
of the distances, that is, if FK be as rfFK, and therefore the area AHIKL
as
p-7 p- ; the attraction of the corpuscle P towards the circle will
PA AH
be as 1 ; that is, as
COR. 2. And universally if the forces of the points at the distances D b(
reciprocally as any power Dn of the distances; that is, if FK be as .
and therefore the area AHIKL as
1 1
" l
PH"
1 PA
, ; the attraction
of the corpuscle P towards the circle will be as
PA"
2
PH"
l
COR. 3. And if the diameter of the circle be increased in itifinitum, and
the number n be greater than unity ; the attraction of the corpuscle P to
wards the whole infinite plane will be reciprocally as PA"
2
, because the
PA
other term vanishes.
PROPOSITION XCI. PROBLEM XLV.
To find the attraction of a corpuscle situate in the axis of a round solid,
to whose several points there tend equal centripetal forces decreasing
in any ratio of the distances whatsoever.
Let the corpuscle P, situate in the axis AB
of the solid DECG, be attracted towards that
solid. Let the solid be cut by any circle as
RFS, perpendicular to the axis ; and in its
semi-diameter FS, in any plane PALKB pass
ing through the axis, let there be taken (by
Prop. XC) the length FK proportional to the
force with which the corpuscle P is attracted
towards that circle. Let the locus of the point
K be the curve line LKI, meeting the planes of the outermost circles AL
and BI in L and I
; and the attraction of the corpuscle P towards the
solid will be as the area LABI. Q..E.I.
COR. 1. Hence if the solid be a cylinder described by the parallelogram
ADEB revolved about the axis AB, and the centripetal forces tending to
the several points be reciprocally as the squares of the distances from the
points ; the attraction of the corpuscle P towards this cylinder will be as
AB PE + PD. For the ordinate FK (by Cor. 1, Prop. XC) will be
PF
as 1 --. The part 1 of this quantity, drawn into the length AB, de
SEC. XIII. OF NATURAL PHILOSOPHY 239
scribes the area 1 X AB ; and the other part
PF
, drawn into the length PB describes the
ix
area 1 into PE AD (as may be easily
shewn from the quadrature of the curve
LKI); and, in like manner, the same part
drawn into the length PA describes the area
L into PD AD. and drawn into AB, the
"At
G
Iv
S
13 M
7J" 1
difference of PB and PA, describes 1 into PE PD, the difference of the
areas. From the first content 1 X AB take away the last content 1 into
PE PD, and there will remain the area LABI equal to 1 into
AB PE -h PD. Therefore the force, being proportional to this area,
is as AB PE + PD.
COR. 2. Hence also is known the force
by which a spheroid AGBC attracts any
body P situate externally in its axis AB.
Let NKRM be a conic section whose ordinate
KR perpendicular to PE may be \
always equal to the length of the line PD,
continually drawn to tlie point D in
which that ordinate cuts the spheroid.
From the vertices A, B, of the spheriod,
let there be erected to its axis AB the perpendiculars AK, BM, respectively
equal to AP. BP, and therefore meeting the conic section in K and M; and
join KM cutting offfrom it the segment KMRK. Let S be the centre of the
spheroid, and SC its greatest semi-diameter : and the force with which the
spheroid attracts the body P will be to the force with which a sphere describ-
, ....,,. ASxCS 2 -PSxKMRK
ed with the diameter AhJ attracts the same body as prrr ^ r-=
1 o -f- Go2 Ao
AS 3
is to fkT^,. And by a calculation founded on the same principles may be
found the forces of the segments of the spheroid.
COR. 3. If the corpuscle be placed within the spheroid and in its axis,
the attraction will be as its distance from the centre. This may be easily
collected from the following reasoning, whether
the particle be in the axis or in any other given
