Imagine several concentric similar
spheres, AB, CD, EF, &c.. the inner
most of which added to the outermost
may compose a matter more dense to
wards the centre, or subducted from
them may leave the same more lax and
rare. Then, by Prop. LXXV, these
sphere? will attract other similar con
SEC. XII.] OF NATURAL PHILOSOPHY. 223
eentric 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 (com
posed of any 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 at
tractive, 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.I).
COR. I. Hence if many spheres of this kind, similar in all respects, at
tract 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 multi
plying 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 peri
odic 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
jf bodies about the foci of conic sections, will take place when an attract
ing sphere, of any form and condition like that above described, is placed
in the focus.
COR. 9. And also when the revolving bodies are also attracting spheres
Df any condition like that above described.
224 THE MATHEMATICAL PRINCIPLES [BOOK I.
PROPOSITION LXXVI1. THEOREM XXXVII.
Tf to 1he several points of spheres there tend centripetal forces propor
tional to the distances of the points from the attracted bodies ; I say,
that the compounded force with which two spheres attract each other
mutually is as the distance between the centres of the spheres.
CASE 1. Let AEBF be a sphere ; S its
centre . P a corpuscle attracted : PASB
the axis of the sphere passing through the
centre of the corpuscle ; EF, ef two planes
cutting the sphere, and perpendicular to
the axis, and equi-distant, 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 centri
petal 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 as 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 that plane drawn into its distance Pg,
or as the equal plane EF drawn into that distance Pg* ; and the sum of the
forces of both planes as the plane EF drawn into the sum of the distances
PG + P^, 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 dis
tance 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, equi-distant 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 distance 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 dis
tance 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
SEC. Xll.] OP NATURAL PHILOSOPHY. 225
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 p be placed within ^- ^\E
the sphere AEBF ; and because the force of the
plane ef upon the corpuscle is as the solid contain
ed under that plane and the distance jog ; and the
contrary force of the plane EF as the solid con
tained under that plane and the distance joG ; 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 distances ;
that is, as that sum drawn into joS, 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 sphere, and as joS, 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 cor
puscles such as jo, 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 joS of
the centres. Q, E.D.
PROPOSITION LXXVIII. THEOREM XXXVIII.
If spheres it* the progress from the centre to the circumference be hoivMtv
dissimilar a->id unequable, but similar on every side round about af all
given distances from the centre ; and the attractive force of evsrt/
point be as the distance of the attracted body ; I say, that the entire
force with which two spheres of this kind attract each other mutitallij
is proportional to the distance between the centres of the spheres.
This is demonstrated from the foregoing Proposition, in the same man
ner as Proposition LXXVI was demonstrated from Proposition LXXY.
COR. Those things that were above demonstrated in Prop. X and LXJV,
of the motion of bodies round the centres of conic sections, take place when
all the attractions are made by the force of sphaerical bodies of the condi
tion above described, and the attracted bodies are spheres of the same kind.
SCHOLIUM.
i have now explained the two principal cases of attractions; to wit,
when the centripetal forces decrease in a duplicate ratio of the distances
r increase in a simple ratio of the distances, causing the bodies in botli
15
226 THE MATHEMATICAL PRINCIPLES [BoOK 1
cases to revolve in conic sections, and composing sphaerical 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 verv
remarkable. It would be tedious to run over the other cases, whose con
clusions are less elegant and important, so particularly as I have done
these. I choose rather to comprehend and determine them all by one gen
eral method as follows.
LEMMA XXIX.
ff about the centre S there be described any circle as AEB, and about the
centre P there be. also described two circles EF, ef, cutting the Jirst in
E and e, and the line PS in F and f
; and there be let fall to PS the
perpendiculars ED, ed ; I say, that if the distance of the arcs EF; ef
be supposed to be infinitely diminished, the last ratio of the evanscent
linr Dd to the evanescent line Ff is the same as that of the line PE to
the live 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, </
!>, DES, it will be as Dd to Ee so
))T to TE, or DE to ES : and because the triangles, Ee?, ESG (by Lem.
VIII, and Cor. 3, Lem. VII) are similar, it will be as Ee to eq or F/soES
to SG ; and, ex ceqno, as Dd to Ff so DE to SG ; that is (because of the
similar triangles PDE; PGS), so is PE to PS. Q.E.D.
PROPOSITION LXXIX. THEOREM XXXIX.
Suppose a superficies as EFfe 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 spherical concavo-convex solid, to
the several equnJ particle* of which there tend equal centripetal forces ;
I soy, that the force with which thit solid attracts a corpuscle situate
in P is in a ratio compounded of the ratio of the solid DE2 X Ff 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 spherical superficies FE which
SEC. xn.j OF NATURAL PHILOSOPHY. 227
is generated by the revolution of the arc FE,
and is cut any where, as in r, by the line</6,
the annular part of the super J cies generated
by the revolution of the arc rE will be as the
lineola Dd, the radius of the sphere PE remainiag
the same; as Archimedes has de
monstrated in his Book of the Sphere and
Cylinder. And the force of this super
ficies exerted in the direction of the lines PE
or Pr situate all round in the conical superficies, will be as this annular
superficies itself; that is as the lineola DC/, or, which is the same, as the
rectangle under the given radius PE of the sphere and the lineola DC/ ; but
that force, exerted in the direction of the line PS tending to the centre S,
will be less in the ratio PI) to PE, and therefore will be as PD X DC/.
Suppose now the line DF to be divided into innumerable little equal par
ticles, each of which call DC/, and then the superficies FE will be divided
into so many equal annuli, whose forces will be as the sum of all the rec
tangles PD X DC/, that is, as |PF 2 - |PD 2
; and therefore as DE-.
Let now the superficies FE be drawn into the altitude F/; and the force
of the solid EF/e exerted upon the corpuscle P will be as DE2 X Ff;
that is, if the force be given which any given particle as Ff exerts upon
the corpuscle P at the distance PF. But if that force be not given, the
force of the solid EF/e will be as the solid DE2 X Ff and that force not
given, conjunctly. Q.E.D.
PROPOSITION LXXX. THEOREM XL.
If to the several equal parts of a sphere ABE described about the centre
S there tend equal centripetal forces ; and from the several points I)
in the axis of the sphere AB in which a corpuscle, as F, 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
DE2 X PS
-, , and as th*force which a particle of the sphere situate in,
the axis exerts at the distance PE upon the corpuscle P conjunctly ; ]
say, that the inhole force with which the, corpuscle P is attracted to
wards 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 DC/, and the whole sphere to be divided into so many sphe
rical concavo-convex laminae EF/e / and erect the perpendicular dn. By
the last Theorem, the force with which the laminas EF/e attracts the cor
puscle P is as DE2 X Ff and the force of one particle exerted at the
228 THE MATHEMATICAL PRINCIPLES [BOOK I.
distance PE or PF, conjunctly.
But (by the last Lemma) Dd is to
F/ as PE to PS, and therefore F/
.
is equal to PE
F/ is equal to Dd X
; and DE2 X
DE2 X PS
PET~ ;
and therefore the force of the la-
DE2 X PS
mina EF/e is as Do? X PT?~
and the force of a particle exerted at the distance PF conjunctly ; that is,
by the supposition, as DN X D(/7 or as the evanescent area DNwrf.
Therefore the forces of all the lamina) exerted upon the corpuscle P are as
all the areas DN//G?, that is, the whole force of the sphere will be as the
whole area ANB. Q.E.D.
COR. 1. Hence if the certripetal force tending to the several particles
p)F 2 vx po
remain always the same at all distances, and DN be made as ;
Jr Jli
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
DE2 X PS
distance of the corpuscle attracted by it, and DN be made as - ^^ ,
the force with which the corpuscle P is attracted by the whole sphere wil]
be as the area ANB.
Cor. 3. Jf 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
T)F 2 y PS
---
. the force with which the corpuscle is 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 ;
DE2 X PS
and D5& be made as ^- ; the force with which a corpuscle is at-
Jr Jtj X
tracted by the whole sphere will be as the area ANB.
PROPOSITION LXXXI. PROBLEM XLI.
T/Le things remaining as above, it is required lo measure the area
ANB.
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. XII, Book II, Elem.) PE2 is equal tf
SEC. XII.] OF NATURAL PHILOSOPHY. 229
PS3 + SE2 + 2PSD. But because
the triangles SPH, SHI are alike,
SE2 or SH2 is equal to the rectan
gle PSI, Therefore PE2 is equal
to the rectangle contained under PS
and PS -f SI + 2SD ; that is, under
PS and 2LS + 2SD ; that is, under
PS and 2LD. Moreover DE2 is
equal to SE2 SD% or SE2
LS 2 + 2SLD LD2
, that is, 2SLD LD 2 ALB. For LSSE2
or LS a SA a
(by Prop. VI, Book II, Elem.) is equal to the rectan
gle ALB. Therefore if instead of DE2 we write 2SLD LD 2 ALB,
the quantity
- -^-, which (by Cor. 4 of the foregoing Prop.) is as PE x
the length of the ordinate DN, will
2SLD x PS LD 2 X PS
now resolve itself into three parts
ALB xPS ...
-TE3rr~ -pfixT" -pE^-v-; whereifinsteadofVwewnt
the inverse ratio of the centripetal force, and instead of PE the mean pro
portional between PS and 2LD, those 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 PE the dis
tance, then 2PS X LD for PE 2
; and DN will become as SL LD
ny |y Suppose DN equal to its double 2SL LD - r^ 5 an<* 2SL
the given part of the ordinate drawn into the length AB will describe the
rectangular area 2SL X AB ; and the indefinite part LD, drawn perpen
dicularly into the same length with a continued motion, in such sort as in
its motion one way or another it may either by increasing or decreasing re-
LB 2 -LA 2
main always equal to the length LD, will describe the area ^ ,
that is, the area SL X AB ; which taken from the former area 2SL X
AB, leaves the area SL X AE. But the third part
-
---, drawn after the
i lit,
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 construction of the Problem. At
the points, L, A, B, erect the perpendiculars L/, Act, B6;
making Aa equal to LB, and Bb equal to LA. Making
L/ and LB asymptotes, describe through the points a, 6,
230 THE MATHEMATICAL PRINCIPLES [BOOK 1
the hyperbolic crrve 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
PE3
thing; as that cube applied to any given plane ; write
2PS X LD for PE2
; and DN will become as
2AS2
SL X AS2
for V, and
AS 2
ALB X AS 2
2PS X LD2
LSI
PS X LD 2PS
that is (because PS, AS, SI are continually proportional), as
ALB X SI
2LD:
LSI
If we draw then these three parts into th
length AB, the first r-pr will generate the area of an hyperbola ; the sec-
L-t \J
, ALB X SI . ALB X SI
ond iSI the area } AB X SI ;
the third
2Ll^ area-2LA
, that is, !AB X SI. From the first subduct the sum of the
2LB
second and third, and there will remain ANB, the area sought. Whence
arises this construction of the problem. At the points L, A, S, B, erect
the perpendiculars L/ Aa Ss, Bb, of which suppose Ss
equal to SI ; and through the point s, to the asymptotes
L/, LB, describe the hyperbola asb meeting the
perpendiculars Aa, Bb, in a and b
; and the rectangle
2ASI, subducted from the hyberbolic area AasbB, will
.. ,, . B 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 parpT^
4 _ tides ; write ~|f- for V, then V 2PS + LD for PE, and DN will become
___
V2SI
X
SI 2 X ALB
2v2SI
X
These three parts drawn into the length AB, produce so many areas, viz.
J-L
2SI 2 X SL . 1
x^ into T r LA
~~~5ot
in* V LB V LA; and
BS1 2 X ALB . "1 1"
VLA 3 v/LB 3
And these after due reduction come
forth __
SEC. XII.] OF NATURAL PHILOSOPHY. 23\.
2SI 3 4 SI 3
~oj-p
And these by subducting the last from the first, become -oT~r
Therefore the entire force with ,7hich the corpuscle P is attracted towards
the centre of the sphere is as-^, that is, reciprocally as PS 3 X PJ
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.
PROPOSITION LXXXIL THEOREM XLI.
In a sphere described about the centre S with the interval SA, if there be
taken SI, SA, SP continually proportional ; ! sat/, 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 tJie centripetal forces tending to the centre in those places P
and I.
