# The Mathematical Principles of Natural Philosophy (1729)/Book 1/Section 13

*Of the attractive forces of bodes which are not of a sphærical figure.*

Proposition LXXXV. Theorem XLII.

*If a body be attracted by another, and its attraction be vastly stronger when it if contiguous to the attracting body, than when they are separated from one another by a very small interval; the forces of the particles of the attracting body decrease, in the recess of the body attracted, in more than a duplicate ratio of the distance of the particles.*

For if the forces decrease ine a duplicate ratio of the distances from the particles, the attraction towards a sphærical body, being (by prop. 74.) recioprocally 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
will 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 concave sphærical 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. 70.) destroyed
by contrary attractions, and therefore have
no effects even in the place of contact. Now if
from these spheres and sphærical orbs we take of
way any parts remote from the place of contact,
and add new parts any where at pleasure; 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. Therefore the
proposition holds good in bodies of all figures.
*Q. E. D.*

*If the forces of the particles of which an attractive body is composed, decrease, in the recess of the attracted body, in a triplicate or more than triplicate ratio of the distance from 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 on interval.*

For that the attraction is infinitely increased
when the attracted corpuscle comes to touch an
attracting sphere of this kind appears by the solution
of problem 41. exhibited in the second and
third examples. The same will also appear (by
comparing those examples and therorem 41. together)
of attractions of bodies concavo-convex
orbs, whether the attracted bodies be
placed without the orbs, or in the cavities whithin
them. And by adding to or taking from those
spheres and orbs, any attractive matter any where
without the place of contact, so that the attractive
bodies may receive any assigned figure, the
proposition will hold good of all bodies universsally. *Q. E. D.*

*if two bodies similar to each other, and consisting of matter equally attractive, attract separately two corpuscles proportional to those bodies, and in a like situation to them; the accelerative attractions of the corpuscle towards the entire bodies will be as* *the accelerative attractions of the corpuscule 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 particle
of one of the bodies to the attraction towards the
correspondent particle in the other body, so are the
attractions towards the several particles of the first
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. D.*

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. As if the
forces of the particles decrease in a duplicate ratio
of the distances from the corpuscles attracted, and
the bodies are as and , and therefore 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 towards the
bodies will be as and , that is, as *A* and *B*
the cubic sides of those bodies. If the forces of
the particles decrease in a triplicate ratio of the
distances from the attracted corpuscles; the accelerative attractions towards the whole bodies will be
as and , that is, equal. If the forces decrease
in a quadruplicate ratio; the attractions towards
the bodies will be as and ; that 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 decrease 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.

*If the attractive forces of the equal particles of any body be at the distance 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*, (Pl. 23. *Fig.* 7.) of
the body *RSTV* attract any corpuscle *Z* with
forces which, supposing 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 so speak) as those particles drawn into
their distances *AZ*, *BZ* respectively. And
let those forces be expressed by the contents under
*A* x *AR*, 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 destroy one other. There remain
then the forces *A* x *GZ* and *B* x *GZ*. These tend
from *Z* towards the centre *G*, and compose the
force ; 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 *C*, and the force of it be compounded
with the force tending to the centre
*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 of
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
infinitum. Therefore the whole force of all the particles of any body whatever *RSTV*, is the same
as if the 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 sphærical; and therefore if that attracting
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.

*If there be several bodies consisting of equal particles whose forces are as the distance 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 proposition.

Cor. Therefore the motion of the attracted body will be the same as if the attracting bodies, preferring 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 proceeds uniformly in a right line; the attracted body will move in an ellipsis having its centre in the common centre of gravity of the attracting bodies.

*If to the several points of any circle there tend equal centripetal forces, increasing or decreasing in any ratio of the distances; it is required to find the force with which a corpuscle it attracted, that is situate any where in a right line which stands at right angles to the plane of the circle at its centre.*

Suppose a circle to be described about the centre
*A (Pl. 24. Fig. 1.) 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 curve meet
the plane of the circle in *L*. In *PA* take *PH*
equal to *PD*, and erect 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* taka *PC* equal
to *Pe*. And because the force with which any
point *E* of the annulus described about the centre
*A* with the interval *AE* in the aforesaid plane, attracts
to it self the body *P*, is supposed to be as
*FK*; and therefore the force with which that point
attracts the body *P* towards *A* is as and
the force with which the whole annulus attracts
the body *P* towards *A*, is as the annulus and
conjunctly; and that annulus also is as
the rectangle under the radius *AE* and the breadth
*Ee*, and this rectangle (because *PE* and *AE*, *Ee*
and *CE* are proportional) is equal to the rectangle
*PE* x *CE* or *PE* x *Ff*; the force with which that
annulus attracts the body *P* towards *A*, will be as
*Pe* x *Ff* and conjunctly; that is as the
content under *Ff* 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 of the distances, that is,
if *FK* be as , and therefore the area *AHIKL* as ; the attraction of the corpuscle *P*
towards the circle will be as ; that is, as .

Cor. 2. And universally if the forces of the
points at the distances *D* be reciprocally as any
power , of the distances; that is, if *FK* be as
, and therefore the area *AHIKL* as ; the attraction of the corpuscle *P*
towards the circle will be as .

cor. 3. And if the diameter of the circle
be increased in infinitum, and the number *n* be
greater than unity; the attraction of the corpuscle
*P* towards the whole infinite plane will be reciprocally
as because the other term
vanishes.

*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 disŧances whatsover.*

Let the corpuscle *P* (Pl. 24. *Fig.* 2.) 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* passing
through the axis. Let there be taken (by prop. 90.)
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* (Pl. 24. *Fig.* 3.)
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. 90.) will be as .
The part *I* of this quantity, drawn into
the length *AB*, describes the area *I* x *AB*; and
the other part
drawn into the length *PB*, describes
the area *I* into (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 *I* into ,
and drawn into *AB*, the difference of *PB* and
*PA* describes *I* into , the difference of the
areas. From the first content *I* x *AB* take away
the last content *I* into , and there will remain
the area *LABI* equal to *I* into .
Therefore the force being proportional to this
area, is as .

Cor. 2. Hence also is known the force by
which a spheroid *AGBC* (Pl. 24. *Fig.* 4.) attracts
any body P situate externally in its axis
*AB*. Let *NKPM* be a conic section whose ordinate
*ER* perpendicular to *PE*, may be always
equal to the length of the line *PD*, continually
drawn to the point *D* in which that ordinate cuts
the spheroid. From the vertices *A*, *B*, of the
spheroid, 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 off from 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
described with the diameter *AB* attracts the same body, as is to
. 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
diameter. Let *AGOF* (Pl. 2.4. FQ. 5.) be an
attracting spheroid, *S* its centre, and *P* the body
attracted. Through the body *P* let there be drawn
the semi-diameter *SPA*, and two right lines *DE*,
*FG* meeting the spheroid in *D* and *E*, *F* and *G*;
and let *PCM*, *HLN* be the superficies of two
interior spheroids similar and concentrical to the
exterior, the first of which passes through the
body *P*, and cuts the right lines *DE*, *FG* in *B*
and *C*; and the latter cuts the same right lines in
*H* and *I*, *K* and *L*. Let the spheroids have all
one common axis, and the parts of the right lines
intercepted on both sides *DP* and *BE*, *FP* and
*CG*, *DH* and *IE*, *FK* and *LG* will be mutually equal;
because the right lines *DE*, *PB*, and
*HI* are bissected in the same points as are also the
right lines *FG*, *PC* and *KL*. Conceive now
*DPF*, *EPG* to represent opposite cones described
with the infinitely small vertical angles *DPF*, *EPG*,
and the lines *DH*, *EI* to be infinitely small also.
Then the particles of the cones *DHKF*, *GLIE*,
cut off by the spheroidical superficies, by reason of
the equality of the lines *DH* and *EI*, will be
to one another as the squares of the distances from the body *P*, and will therefore attract that
corpuscle equally. And by a like reasoning if the
spaces *DPF*, *EGCB* be divided into particles by
the superficies of innumerable similar spheroids concentric
to the former and having one common
axis, all these particles will equally attract on both
sides the body *P* towards contrary parts. Therefore
the forces of the cone *DPF*, and of the conic
segment *EGCB* are equal and by their contrariety
destroy each other. And the case is the
same of the forces of all the matter that lies without
the interior spheroid *PCBM*. Therefore the
body *P* is attracted by the interior spheroid *PCBM*
alone, and therefore (by cor. 3. prop. 71.) its attraction
is to the force with which the body *A*
is attracted by the whole spheroid *AGOD*, as the
distance *PS* to the distance *AS*. *Q. E. D.*

*An attracting body being given, it is required to find the ratio of the decrease of the centripetal forces tending to its several points.*

The body given must be formed into a sphere,
a cylinder, or some regular figure whose, law of
attraction answering to any ratio of decrease may
be found by prop. 80. 81 and 91. Then, by experiments,
the force of the attractions must be
found at several distances, and the law of attraction
towards the whole, made known by that means,
will give the ratio of the decrease of the forces of
the several parts; which was to be found.

*If a solid be plane one one side, and infinitely extended on all other sides, and consist of equal particles equally attractive, whose forces decrease, in the recess from the solid, in the ratio of any power greater than the square of the disŧances; and a corpuscle placed towards either part of the plane is attracted by the force of the whole solid; I say that the attractive force of the whole solid, in the attractive force of the whole solid, in the recess from its plane superficies, will decrease in the ratio of a power whose side is the distance of the corpuscle from the plane, and its index less by* 3 *than the index of the power of the distance.*

Case. 1. Let *LGI* (Pl. 24. *Fig.* 6.) be the
plane by which the solid is terminated. Let the
solid lie on that hand of the plane that is towards
*I*, and let it be resolved into innumerable planes
*mHM*, *nIN*, *oKO*, &c. parallel to *GL*. And
first let the attracted body *C* be placed without
the solid. Let there be drawn *CGHI* perpendicular
to those innumerable planes, and let the attractive
forces of the points of the solid decrease
in the ratio of a power of the distances whose index
is the number as not less than 3. Therefore (by cor. 3. prop. 90.) the force with which any
plane *mHM* attracts the point *C*, is reciprocally
as . In the plane *mHM* take the length
*HM* reciprocally proportional to , and
that force will be as *HM*. In like manner in the
several planes *IGL*, *nIN*, *oKO*, &c. take the lengths
*GL*, *IN*, *KO*, &c. reciprocally proportional to
, , , &c. and the forces
of those planes will be as the lengths so taken,
and therefore the sum of the forces as the sum of
the lengths, that is, the force of the whole solid
as the area *GLOK* produced infinitely towards *OK*.
But that area (by the known methods of quadratures)
is reciprocally as , and therefore the
force of the whole solid is reciprocally as . *Q. E. D.*

Case 2. Let the corpuscle (*Fig. 7.*) be now placed
on that hand of the plane IGL that is within the solid,
and take the distance *CK* equal to the distance
*CG*. And the part of the solid *LGI* x *KO* terminated
by the parallel planes *IGL*, *oKO*, will attract
the corpuscle, situate in the middle, neither
one way nor another, the contrary actions of the
opposite points destroying one another by reason
of their equality. Therefore the corpuscle *C* is attracted
by the force only of the solid situate beyond
the plane *OK*. But this force (by case 1.) is reciprocally
as , that is (because *CG*, *CK* are
equal) reciprocally as . *Q. E. D.*

Cor. 1. Hence if the solid *LGIN* be terminated
on each side by two infinite parallel planes *LG*,
*IN*; its attractive force is known, subducting from
the attractive force of the whole infinite solid *LGKO*,
the attractive force of the more distant part *NIKO*
infinitely produced towards *KO*.

Cor. 2. If the more distrant part of this solid be rejected, because its attraction compared with the attraction of the nearer part is inconsiderable; the attraction of that nearer part will, as the distance increases, decrease nearly in the ratio of the power

Cor. 3. And hence if an finite body, plane on one side, attract a corpuscle situate over-against the middle of that plane, and the distance between the corpuscle and the plane compared with the dimensions of the attracting body be extremely small; and the attracting body consist of homogeneous particles, whose attractive forces decrease in the ratio of any power of the distances greater than the quadruplicate; the attractive force of the whole body will decrease very nearly in the ratio of a power whose side is that very small distance, and the index less by 3 than the index of the former power. This assertion does not hold good however of a body consisting of particles whose attractive forces decrease in the ratio of the triplicate power of the distances; because in that cases the attraction of the remoter part of the infinite body in the second corollary is always infinitely greater than the attraction of the nearer part.

If a body is attracted perpendicularly towards a given plane, and from the law of attraction given the motion of the body be required; the problem will be solved by seeking (by prop. 39.) the motion of the body descending in a right line towards that plane, and (by cor. 2. of the laws) compounding that motion with an uniform motion, performed in

the direction of lines parallel to that plane. And on the contrary if there be required the law of the attraction tending towards the plane in perpendicular directions, by which the body may be caused to move in any given curve line, the problem will be solved by working after the manner of the third problem.

But the operations may be contracted by resolving
the ordinates into converging series. As if to a
base *A* the length *B* be ordinately applied in any
given angle, that length be as any power of
the base ; and there be sought the force with
which a body, either attracted towards the base or
driven from it in the direction of that ordinate,
may be caused to move in the curve line which
that ordinate always describes with its superior
extremity; I suppose the base to be increased by a
very small part *O*, and I resolve the ordinate
into an infinite series &c. and I suppose the
force proportional to the term of this series in which
O is of two dimensions, that is to the term
. Therefore the force sought is as
, or,
which is the same thing, as . As if the ordinate describe a
parabola, m begin = 2, and n = 1, the force will
be as the given quantity , and therefore is . given. Therefore with a given force the body will move in a a parabola, as *Galileo* has demonstrated. If the ordinate describe an hyperbola, *m* being=0—1, and *n*=1; the force will be as or ; and therefore a force which is as the cube of the ordinate will cause the body to move in an hyperbola. But leaving this kind of propositions, I
shall go on to some others relating to motion which I have not yet touched upon.