# The Mathematical Principles of Natural Philosophy (1729)/Book 1/Section 12

*Of the attractive forces of sphærical bodies.*

Proposition LXX. Theorem XXX.

*If to every point of a sphærical surface there tend equal centripetal forces decreasing in the duplicate ratio of the disstances from those points; I say that a corpuscle placed within that superficies will not be attracted by those forces any way.*

Let *HIKL* (*Pl*. 21. *Fig*. 4.) be that sphærical superficies, and *P* a corpuscle placed within. Through *P* let there be drawn to this superficies the two lines *HK*, *IL*, intercepting very small arcs *HI*, *KL*; and because (by cor. 3. lem. 7.) 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 sphærical 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 ratio's 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 sphærical superficies are destroyed
by contrary attractions. Therefore the body
*P* will not be any way impelled by those attractions.
*Q. E. D.*

*The same things supposed as above, I say that a corpuscle placed without the sphærical superficies is attracted towards the centre of the sphere with a force reciprocally proportional to the square of its distance from that centre.*

Let *AHKB*, *ahkb* (Pl. 21. *Fig.* 5.) be two
equal sphærical superficies described about the centres
*S*, *s*; their diameters *AB*, *ab*; and let *P* and
*p* be two corpuscles situate without the spheres 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*, *bk*, *IL*, *il*; and to those lines let
fall the perpendiculars *SD*, *sd*, *SE*, *se*, *IR*, *ir*;
of which let *SD*, *sd* cut *PL*, *pl* in *F* and *f*.
Let fall also to the diameters the perpendiculars
*IQ*, *iq*. Let now the angles *DPE*, *dpe* vanish;
and because *DS* and *ds*, *ES* and *es* are equal, the
lines *PE*, *PF*, and *pe*, *pf*, and the lineolæ *DF*,
*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 æquo*, as
*PI* x *pf* to *PF* x *pi* so is *RI* to *ri*, that is (by cor.
3. lem. 7.) so is the arc *IH* to the arc *ih*. Again
*PI* is to *PS* as *IQ* to *SE*, and *ps* ro *pi* as *se* or *SE* to *iq*; and *ex æquo* *PI* x *ps* to *PS* x *pi* as *IQ*
to *iq*. And compounding the ratio's is to , as *IH* x *IQ* to *ib* x *iq*; that
is, as the circular superficies which is described by
the arc *IH* as the semicircle *AKB* revolves about
the diameter *AB*, is to the circular superficies
described by the arch *ih* as the semicircle *akb* revolves
about the diameter *ab*. And the forces
with which these superficies attracts 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* x *PS*. And these
forces again are to the oblique parts of them
which (by the revolution of forces as in cor. 2.
of the laws) tend to the centres in the directions
of the lines *PS*, *ps*, as *PI* to *PQ* and *pi* to *pq*; that is (because of the like triangles *PIQ* and *PSF*, *piq* and *psf*) as *PS* to *PF* and *ps* to *pf*. Thence *ex equo*, the attraction of the corpuscle *P* towards *S* is to the attraction
of the corpuscle *p* towards *s*, as
is to , that is, as to . And by a like reasoning the forces with which the superficies described by the revolution of the arcs *KL*, *kl*attract those corpuscles, will be as to . And in the same ratio will be the forces of all the circular superficies into which each of the sphærical 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 sphærical superficies exerted upon those corpuscles will be it: the same ratio. *Q. E. D.*

Proposition LXXII. Theorem XXXII.

*If to the sveral points: of a sphere there tend equal centripetal forces decreasing 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 diameter of the sphere to the distance of the corpuscle from its centre; I say that the force with which the corpuscle is * *attracted is proportional 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 several 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 disŧances inversely.
But the particles are as the spheres, that
is in a triplicate ratio 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. 5.
prop. 4.

Cor. 3. If to the several points of any two
solids whatever, of like figure 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.*

*If to the sevaral points of a given sphere there tend equal centripetal forces decreasing in a duplicate ratio of the disŧances from the points; I say that a corpuscle placed within sphere is attracted by a force proportional to its disŧance from the centre.*

In the sphere *ABCD* (Pl. 21. *Fig.* 6.) described
about the centre *S*, let there be placed the
corpuscle *P*; and about the same centre *S*, with
the interval *SP*, conceive described an interior
sphere *PEQF*. It is plain (by prop. 70.) that
the concentric sphærical supericies of which the
difference *AEBF* of the spheres is composed,
have no effect at all upon the body *P*; their attractions
being destroyed by contrary attractions.
There remains therefore only the attraction of the
interior sphere *PEQF* And (by prop. 72.) this
is as the distance *PS*. *Q. E. D.*

By the superficies of which I here imagine the solidg composed, I do not mean superficies purely

mathematical, but orbs so extreamly 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 incossiderable.

*The same things supposed, I say that a corpuscle situate without a force reciprocally proportional to the square of its distance form the centre.*

For suppose the sphere to be divided into innumerable
concentric sphærical superficies, and the attractions
of the corpuscle arising from the several
superficies will be reciprocally proportional to the
square of the distance of the corpuscle from the
centre of the sphere (by prop. 71.) 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. 72.) 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 attracted 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.

*If to the several points of a given sphere there tend equal centripetal force; decresing in a duplicate ratio of the distances from the points; I say that another similar sphere will be attracted by it with a force reciprocal proportional to the square of the disŧance of the centres.*

For the attraction of every particle is reciprocally
as the square of its distance from the centre of the
attracting sphere (by prop. 74.) and is therefore
the same as if that whole attracting force issued from one single 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 it self attracted by the several
particles of the attracted sphere with the same
force with which they are attracted by it. But
that attraction of the corpuscle would be (by prop.
74.) reciprocally proportional 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 attract. 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 3.) the attracted 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.

*If spheres be however dissimilar (as to density of matter and attractive force) in the progress right onward from the centre to the circumference; but every where similar, at every given disŧance 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 other, will be reciprocally proportional to the force of the distance of the centres.*

Imagine several concentric similar spheres, *AB*,
*CD*, *EF*, &c. (Pl. 22. *Fig.* 1.) the innermost of
which added to the outermost may compose a
matter more dense towards the centre, or subducted
from them may leave the same more lax and rare.
Then by prop. 75. these spheres will attract other
similar concentric spheres *GH*, *IK*, *LM*, &c, each
the other, with forces reciprocally proportional to
the square of the distance *SP*. And by composition
or division, the sum of all those forces, or the
excess of any of them above the others; that is, the
entire force with which the whole sphere *AB*
(composed of an concentric spheres or of their differences) will attract the whole sphere *GH* (composed of any concentric spheres or their differences) in the same ratio. Let the number of the concentric spheres be increased in infinitum, so that the density of the matter together with the attractive force may, in the progress from the circumference to the centre, increase or decrease according to any given law; and by the addition of matter not attractive let the deficient density be supplied that so the spheres may acquire any form desired; and the force with which one of these attracts the other, will be still, by the former reasoning, in the same ratio of the square of the distance inversely. *Q. E. D.*

Cor. 1. Hence if many spheres of this kind, similar in all respects, attract each other mutually; the accelerative attractions of each to each, at any equal distances of the centres, will be as the attracting spheres.

Cor. 2. And at any unequal distances, as the attracting spheres applied to the squares of the distances between the centres.

Cor. 3. The motive attractions, or the weights of the spheres towards one another will be at equal distances of the centres as the attracting and attracted spheres conjunctly; that is, as the products arising from multiplying the spheres into each other.

Cor. 4. And at unequal distances, as those products directly and the squares of the distances between the centres inversely.

Cor. 5. These proportions take place also, when the attraction arises from the attractive virtue of both spheres mutually exerted upon each other. For the attraction is only doubled by the conjunction of the forces, the proportions remaining as before.

Cor. 6. If spheres of this kind revolve about others at rest, each about each; and the distances between the centres of the quiescent and revolving bodies are proportional to the diameters of the quiescent bodies; the periodic times will be equal.

Cor. 7. And again, if the periodic times are equal, the distances will be proportional to the diameters.

Cor. 8. All those truths above demonstrated, relating to the motions of bodies about the foci of conic sections, will take place, when an attracting sphere, of any form and condition like that above described, is placed in the focus.

Cor. 9. 9. And also when the revolving bodies are also attracting spheres of any condition like that above described.

*If to the several points of spheres there tend centripetal forces proportional to the disŧances of the points from the attracted bodies; I say that the compounded force with which two spheres attract each other mutually is or the disŧance between the centres of the spberes.*

Case 1. Let *AEBF* (Pl. 22. *Fig.* 2.) be a
sphere; *S* its centre; *P* a corpuscle attracted;
*PASB* the axis of the sphere pulling through the
centre of the corpuscle; *EF*, *ef* two planes cutting
the sphere, and perpendicular to the axis, and equidistant, one on one side, the other on the other,
from the centre of the sphere; *G* and *g* the
intersections of the planes and the axis; and *H* any
point in the plane *EF*. The centripetal force of
the point *H* upon the corpuscle *P*, exerted in the
direction of the line *PH* is as the distance *PH*;
and (by cor. 2. of the laws) the same exerted in the
direction of the line *PG*, or towards the centre *S*,
is at the length *PG*. Therefore the force of all
the points in the plane *EF* (that is of that whole
plane) by which the corpuscle *P* is attracted towards
the centre *S* is as the distance *PG* multiplied
by the number of those points, that is as the
solid contained under that plane *EF* and the distance
*PG*. And in like manner the force of the
plane *ef' by which the corpuscle *P* is attracted towards*
the centre *S*, is as at plane drawn into its
distance *Pg*, or as the equal plane *EF* drawn into
that distance *P*; and the sum of the forces of
both planes as are plane *EF* drawn into the sum
of the distances *PG* + *Pg*, that is as that plane
drawn into twice the distance *PS* of the centre
and the corpuscle; that is, as twice the plane *EF*
drawn into the distance *PS*, or as the sum of the
equal planes *EF* + *ef* drawn into the same distance.
And by a like reasoning the forces of all the
planes in the whole sphere, equidistant on each side
from the centre of the sphere, are as the sum of
those planes drawn into the distance *PS*, that
is, as the whole sphere and the disŧance *PS* conjunctly.
*Q. E. D.*

Case 2. Let now the corpuscle *P* attract the sphere
*AEBF*. And by the same reasoning it will appear
that the force with which the sphere is attracted is as
the distance *PS*. *Q. E. D.*

Case 3. Imagine another sphere composed of
innumerable corpuscles *P*; and because the force
with which every corpuscle is attracted is as the
distance of the corpuscle from the centre of the
first sphere, and as the same sphere conjunctly, and
is therefore the same as if it all proceeded from a
single corpuscle situate in the centre of the sphere;
the entire force with which all the corpuscles in the
second sphere are attracted, that is, with which that
whole sphere is attracted, will be the same as if that
sphere were attracted by a force issuing from a
single corpuscle in the centre of the first sphere; and is
therefore proportional to the distance between the centres
of the spheres. *Q. E. D.*

Case 4. Let the spheres attract each other mutually,
and the force will be doubled. but the proportion
will remain. *Q. E. D.*

Case 5. Let the corpuscle be placed within the
sphere *AEBF*; (*Fig.* 3.) and because the force of the
plane *ef* upon the corpuscle is as the solid contained
under that plane and the distance *pg*; and the contrary
force of the plane *EF* as the solid contained
under that plane and the distance *pG*; the force
compounded of both will be as the difference of
the solids, that is as the sum of the equal planes
drawn into half the difference of the distance that
is, as that sum drawn into *PS*, the distance
of the corpuscle from the centre of the sphere.
And by a like reasoning, the attraction of all the
planes *EF*, *ef* throughout the whole sphere, that
is, the attraction of the whole sphere, is conjunctly
as the sum of all the planes, or as the whole spheres
and as *pS*, the distance of the corpuscle from the
centre of the sphere. *Q. E. D.*

Case 6. And if there be composed a new sphere
out of innumerable corpuscles such as *p*, situate
within the first sphere *AEBF*; it may be proved
as before that the attraction whether, single of one
sphere towards the other, or mutual of both towards
each other, will be as the distance *pS* of the centres.
*Q. E. D.*

*If spheres in the progression from the centre to the circumference be however dissimimar and unequable, but similar on every side round about at all given disŧances from the centre; and the attractive force of every point be as the disŧance of the attracted body; I say that the entire force with which two spheres of this kind attract each other mutually is proportional to the centres of the spheres.*

This is demonstrated from the foregoing proposition in the same manner as the 76th proposition was demonstrated from the 75th

Cor. Those things that were above demonstrated in prop. 10. and 64. of the motion of bodies round the centres of conic sections, take place when all the attractions are made by the force of sphærical bodies of the condition above described, and the attracted bodies are spheres of the same kind.

I have now explained the two principal case of attractions; to wit, when the centripetal forces decrease in a duplicate ratio of the distances, or increase in a simple ratio of the distances; causing the bodies in both cases to revolve in conic sections, and composing sphærical bodies whose centripetal forces observe the same law of increase or decrease in the recess from the centre as the forces of the particles themselves do; which is very remarkable. It would be tedious to run over the other cases, whose conclusions are less elegant and important, so particularly as I have done these. I chuse rather to comprehend and determine them all by one general method as follows.

*If about the centre* S (Pl. 22. Fig. 4.) *these le described arty circle at* AEB, *and about the centre* P *there be also described two circles* EF, *ef*, cutting the first in* R *and* e, *and the line* PS *in* F *and* f; *and the line* PS *in* F *and* f; and there be let fall to* PS *the perpendiculars* ED, ed; *I say, that, if the disŧance of the arcs* EF, ef, *be supposed to be infinitely, the last ratio of the evanescent *evanescent line* Dd *to the evanescent line* Ff *is the same as that of the line* PE *to the line* PS.*

For if the line *Pe* cut the arc *EF* in *q*; and
the right line *Ee*, which coincides with the
evanescent arc *Ee*, be produced and meet the right
line *PS* in *T*; and there be let fall from *S* to
*PE* the perpendicular *SG*; then because of the
like triangles *DTE*, *dTe*, *DES*; it will be as
*Dd* to *Ee* so *DT* to *TE*, or *DE* to *ES*; and
because the triangles *Eeq*, *ESG* (by lem. 8. and
cor. 3. lem. 7.) are similar, it will be as *Ee* to
*eq* or *Ff* so *ES* to *SG*; and *ex æquo*, as
*Dd* to *Ff* so *DE* to *SG*; that is (because is
the similar triangles *PDE*, *PGS*) so is *PE* to
*PS*. *Q. E. D.*

*Suposse a superficies as* EFfe (Pl. 22 Fig. 5.) *to have its breadth infinitely diminished, and to be just vanishing; and that the same superficies by its revolution round the axis* PS *describes a sphærical concavo-convex solid to the several equal particles of which there tend equal centripetal forces; I say that the force with which that solid attracts a corpuscle situate in* P, *is in a ratio compunded of the ratio of the solid* *and the ratio of the force with *which the given particle in the place* Ff *would attract the same corpuscle.

For if we consider first the force of the sphærical
superficies *FE* which is generated by the revolution
of the arc *FE*, and is cut any where, as
in *r*, by the line *de*; the annular part of the superficies
generated by the revolution of the arc *rE*
will be as the lineola *Dd*, the radius of the sphere
*PE* remaining the same; as *Archimedes* has demonstrated
in his book of the sphere and cylinder.
And the force of this superficies exerted in the
direction of the lines *PE* or *Pr* situate all round
in the conical superficies, will be as this annular
superficies it self; that is as the lineola *Dd*, or
which is the same as the rectangle under the given
radius *PE* of the sphere and the lineola *Dd*; but
that force, exerted in the direction of the line *PS*
tending to the centre *S*, will be less in the ratio
of *PD* to *PE*, and therefore will be as *FD* x *Dd*.
Suppose now the line *DF* to be divided into innumerable
little equal particles, each of which call
*Dd*; and then the superficies *FE* will be divided
into so many equal annuli, whose forces will be as
the sum of all the rectangles *PD* x '*Dd*, that is, as
, and therefore as . Let
now the superficies *FE* be drawn into the altitude
*Ff*; and the force of the solid *EFfe* exerted
upon the corpuscle *P* will be as ; that
is, if the force be given which any given particle
as *Ff* exerts upon the corpuscle *P* at the disŧance
*PF*. But if that forte be not given, the force of
the solid *EFfe* will be as the solid
and that force not given, conjunctly. *Q. E. D.*

*If to the several equal parts of a sphere* ABE, (Pl. 22. Fig. 6.) *described about the centre* S, *there tend equal centripetal forces; and from the several points* D *in the axis of the sphere* AB *in which a corpuscle, as, is placed, there be erected the perpendiculars* DE *meeting the sphere in* E, *and if in those perpendiculars the lengths* DN *be taken as the quantity* *and as the force which a particle of the sphere situate in the axis exerts at the distance* PE upon the corpuscle* P, *conjunctly; I say that the whole force with which the corpuscle* P is attracted towards the sphere is as the area* ANB, *comprehended under the axis of the sphere* AB, *and the curve line* ANB, *the locus of the point* N.

For supposing the construction in the last lemma
and theorem to stand. conceive the axis of the
sphere *AB* to be divided into innumerable equal
particles *Dd*, and the whole sphere to be divided
into so many sphærical concavo-convex laminæ '*EFfe*; and erect the perpendicular *dn*. By the last theorem
the force with which the laminæ *EFfe* attracts
the corpuscle *P*. is as and the force
of one particle exerted at the distance *PE* or *PF*,
conjunctly. But (by the last lemma) *Dd* is to
*Ff* as *PE* to *PS*, and therefore *Ff* is equal to
and is equal to and
therefore the force of the laminæ *EFfe* and the force of particle exerted
at the disŧance *PF* conjunctly; that is supposition, as *DN* x *Dd*, or as the evanescent area *DNnd*. Therefore the forces of all the laminæ exerted upon the corpuscle *P* are as all the areas *DNnd*, that is, the whole force of the sphere will be as the whole area *ANB*. *Q. E. D.*

Cor. 1. Hence if the centripetal force tending
to the several particles remain always the sæme at
all distances, and *DN* be made as the whole
force with which the corpuscle is attracted by the
sphere is as the area *ANB*.

Cor. 2. If the centripetal force of the particles
be, reciprocally as the distance of the corpuscle
attracted by it, and *DN* be made as
the force with which the corpuscle B is attracted
by the whole sphere will be as the area
*ANB*.

Cor. 3. If the centripetal force of the particles
be reciprocally as the cube of the distance of the
corpuscle attracted by it, and *DN* be made as

the force with which the corpuscle
attracted by the whole sphere will be as the area
*ANB*.

Cor. 4. And universally if the centripetal force
tending to the several particles of the sphere be
supposed to be reciprocally as the quantity *V*; and *DN* be made as ; the force with which
5 corpuscle is attracted by the whole sphere will be as
the area *ANB*.

*The things remaining as above it is required to measure the area* ANB. (Pl. 23. Fig. 1.)

From the point *P* let there be drawn the right
line *PH* touching the sphere in *H*; and to the
axis *PAB* letting fall the perpendicular *HI*, bisect
*PI* in *L*; and (by prop. 12. book 2. elem.) is equal to . But because
the triangles *SPH*, *SHI* are alike. or
is equal to the rectangle *PSI*. Therefore
is equal to the rectangle contained under *PS* and
*PS* + *SI* + 2'SD*; that is under *PS* and 2*LD*. Moreover is equal to , or , that is, . For or (by prop. 6. book 2. elem.) is equal*
to the rectangle *ALB*. Therefore if instead
of we write ; the quantity , which (by cor. 4. of the
foregoing prop.) is as the length of the ordinate
*DN* will now resolve it self into three parts
; where if instead of *V* we write the inverse
ratio of the centripetal force, and instead of *PE* the mean
proportional between *PS* and 2*LD*; thos three
parts will become ordinates to so many curve
lines, whose areas are discovered by the common
methods. *Q. E. D.*

Example 1. If the centripetal force tending to the
several particles of the sphere be reciprocally as the
distance; instead of *V* write* P*E the distance;
then . Suppose *DN* equal
to its double ;
and 2*SL* the given part of the ordinate drawn into the
length *AB* will describe the rectangular area
2*SL* x *AB*; and the indefinite part *LD*, drawn
perpendicularly into the same length with a
continued motion, in such fort as in its motion
one way or another it may either by increasing
or decreasing remain always equal to the length
*LD*, will desrive that is, the area *SL* x *AB*; which taken from the former
area 2*SL* x *AB* leaves the area *SL* x *AB*. But
the third part , drawn after the same manner with a continued motion perpendicularly into the
same length, will describe the area of an hyperbola,
which subducted from the area *SL* x *AB* will
leave *ANB* the area sought. Whence arises this

*L*,

*A*,

*B*(

*Fig*. 2.) erect the perpendiculars

*Ll*,

*Aa*,

*Bb*;

making *Aa* equal to *Ll*, and *Bb* equal to *LA*.
Making *Ll*, and *LB* asymptotes, describe through
the points *LA*, the hyperbolic curve *ab*. And
the chord *ba* being drawn will inclose the area *aba*
equal to the area sought *ANB*.

Example 2. If the centripetal force tending to
the several particles of the sphere be reciprocally as
the cube of the distance, or (which is the same
thing) as that cube applied to any given plane;
write for *V*, and 2*PS* x *LD* for ; and
*DN* will become as that is (because *PS*, *AS*, *SI' are*
continually proportional) as .
If we draw then these three parts into the length
*AB*, the first will generate the area of an
hyperbole; the second , the area ;
the third ,
the area that is . Form the first subduct the
sum of the second and third, and there will remain
*ANB* the area sought. Whence arisfes this

*L*, *A*, *S3*, *B*, (*Fig.* 3.) erect the perpendicualrs *Ll*, *Aa*, *Ss*, *Bb*, of which suppose *Ss* equal to *SI*;
and through the point *s*, to the asymptotes *Ll*,
*LB*, describe the hyperbola *asb* meeting the perpendiculars
*Aa*, *Bb*, in *a* and *b*; and the rectangle
2*ASI* subducted from the hyperbola *AasbB*,
will leave *ANB* the area sought.

Example 3. If the centripetal force tending to
the several particles of the spheres decrease in a
quadruplicate ratio of the distance from the particles;
write for *V*, then for *PE*, and
*DN* will become as
These three parts drawn into the length *AB*, produce
so many areas, viz. into ; into ; and into . And these after due reduction come forth , and . And
these by subducting the last from the first become
. Therefore the entire force with which
the corpuscle *P* is attracted towards the centre of
the sphere is as , that us reciprocally as . *Q. E. I.*

By the same method one may determine the attraction of a corpuscle situate within the sphere, but more expeditiously by the following theorem.

*In a sphere described about the centre* S (Pl. 23. Fig. 4.) *with the interval* SA, *if there be taken* SI, SA, SP *continually proportional; I say that the attraction of a corpuscle that the attraction of a corpuscle within the sphere in any place* I, *is to its attraction without the sphere in the place *P*, in a ratio compounded of the subduplicate ratio of* IS, PS the distances from the centre, and the subduplicate ratio of the centripetal forces tending to the centre in the places* P *and* I. *

As if the centripetal forces of the particles of
the sphere be reciprocally as the distances of the
corpuscle attracted 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 subduplicate
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
ratio's 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 distance, it
will be found that the attraction in *I* is to the
attraction in *P* as the disŧance *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
to ; if in a quadruplicate ratio as to .
Therefore since the attraction in P was found in
this last case to be reciprocally as , the
attraction in *I* will be reciprocally as ,
that is, because is given, reciprocally as .
And the progression is the same in infinitum.
The demonstration of this theorem is as follows.

The things remaining as above constructed and
a corpuscle being in any place *P*, the ordinate *DN*
was found to be as . Therefore if *IE*
be drawn, that ordinate For any other place of
the corpuscle as *I*, will become (*mutatis mutandis*)
as . Suppose the centripetal forces flowing
from any point of the sphere as *E*, to be
to each other at the disŧances *IE* and *PE*, as
to , (where the number in *n* denotes the
index of the powers of *PE* and *IE*) and those ordinates
will become as and whose ratio to each other is as to
.
Because *SI*, *SE*, *SP* are in
continued proportion, the triangles *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 to . But
the ratio of *PS* to *SA* is subduplicate of that of
the distances *PS*, *SI*; and the ratio of to
(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 consequently the areas which the
ordinates describe, and the attractions proportional to
them, are in a ratio compounded of those subduplicate
ratio's. *Q. E. D.*

*To find the force with which a corpuscle placed in the centre of sphere is attracted towards any segment of that sphere whatsover.*

Let P (Pl. 23. Fig. 5.) be a body in the centre
of that sphere, and *RBSD* a segment thereof
contained under the plane *RDS* and the sphærical
superficies *RBS*. Let *DB* be cut in *F* by a sphærical
superficies *EFG* described from the centre *P*,
and let the segment be divided into the parts
*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 *Archimedes* has demonstrated) that superficies
will he as *PF* x *DF* x *O*. Let us suppose
besides the attractive forces of the particles of
the sphere to be reciprocally as that power of the
distances, of which *n* is index; and the force with
which the superhcies *EFG* attracts the body *P*,
will be (by prop. 79.) as, that is, as
. Let the perpendicular
*FN*, drawn into *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.*

*To find the force with which a corpuscle, placed without the centre of a sphere in the axis of any segment, is attracted, by that segment.*

Let the body *P* placed in the axis *ADB* of the segment
*EBK* (Pl. 23. *Fig.* 6.) be attracted by that
segment. About the centre *P* with the interval *PE*
let the sphærical 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. 81. and the force of the latter part by
prop. 83. and the sum of the forces will be the force
*Pf* the whole segment *EBKDE*. *Q. E. I.*

The attractions of sphærical 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 necessary to my design. It will be
sufficient to subjoin some general propositions 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 enquiries.