diameter. Let AGOF be an attracting sphe
roid, 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,
FC meeting the spheroid in 1) and E, F and
G ; and let PCM, HLN be the superficies of
240 THE MATHEMATICAL PRINCIPLE* ffioOK 1.
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 ; arid the latter cuts the same right lines in H and I, K and L.
I ,et 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 bisected in the same point, as are also the right lines FG, PC, and KL.
Conceive now DPF. EPG to represent opposite cones described with the
infmitely small vertical angles DPF, EPG, and the lines DH, El 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 ET; 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 rea
soning if the spaces DPF, EGCB be divided into particles by the superfi
cies of innumerable similar spheroids concentric to the former and having J. O
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 de
stroy 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. 1 ,XXII) its attraction is to the force with which the body A is at
tracted by the whole spheroid AGOD as the distance PS to the distance
AS. Q.E.D.
PROPOSITION XCII. PROBLEM XLVI.
An attracting body being given, it is required to find the ratio of the de
crease of the centripetalforces tending to its several points.
The body given must be formed into a sphere, a cylinder, or some regu
lar figure, whose law of attraction answering to any ratio of decrease may
be found by Prop. LXXX, LXXXI, and XCI. 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.
PROPOSITION XCIII. THEOREM XLVII.
If a solid be plane on one side, and infinitely extended on all otljer sides,
and consist of equal particles equally attractive, ivhose forces decrease,
in the recessfrom the solid, in the ratio of any power greater than the
square of the distances ; and a corpuscle placed towards eitfar part of
the plane is attracted by the force of the whole solid ; I say that tfie
attractive force of the whole solid, in the recessfrom its platw superfi
XIILj OF NATURAL PHILOSOPHY". 241
n
H
m
G
ties, will decrease in the ratio of a power whose side is the distance oj
the corpuscle from the plane, and its index less by 3 than the index oj
the power of the distances.
CASE 1. Let LG/be the plane by which
the solid is terminated. Let the solid
lie on that hand of the plane that is to
wards I, and let it be resolved into in- _.
numerable planes mHM, //IN, oKO,
(fee., parallel to GL. And first let the
attracted body C be placed without the
solid. Let there be drawn CGHI per
pendicular 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 n not less than 3.
Therefore (by Cor. 3, Prop. XC) the force with which any plane mHM
attracts the point C is reciprocally as CHn 2
. In the plane mHM take the
length HM reciprocally proportional to CH 1 2
, and that force will be as
HM. In like manner in the several planes /GL, //,TN, oKO, (fee., take the
lengths GL, IN, KO, (fee., reciprocally proportional to CGn 2
, CI 1 2
,
CKn 2
, (fee., and the forces of those planes will bs as the lengths so taken,
and therefore the sum of the forces as the sum of the lengths, that is, the
force of the Avhole solid as the area GLOK produced infinitely towards
OK. But that area (by the known methods of quadratures) is reciprocally
as CGn 3
, and therefore the force of the whole solid is reciprocally as
CG"-
3
. Q.E.D.
CASE 2. Let ttecorpuscleC be now placed on that
hand of the plane /GL that is within the solid,
and take the distance CK equal to the distance
CG. And the part of the solid LG/oKO termi
nated by the parallel planes /GL, oKO, will at
tract the corpuscle C, 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 CKn 3
, that is, (because CG, CK are equal) reciprocally as
CG"
3
. Q,.E.D.
COR. 1. Hence if the solid LGIN be terminated on each sitfe by two in
finite parallel places 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 distant part of this solid be rejected, because its at
traction compared with the attraction of the nearer part is inconsiderable,
16
242 THE MATHEMATICAL PRINCIPLES [BOOK 1
the attraction of that nearer part will, as the distance increases, decrease
nearly in the ratio of the power CGn 3
.
Con. 3. And hence if any finite body, plane on one side, attract a cor
puscle 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 case, 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.
SCHOLIUM.
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 Pro
blem will be solved by seeking (by Prop. XXXIX) 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 direc
tion of lines parallel to that plane. And, on the contrary, if there be re
quired the law of the attraction tending towards the plane in perpendicu
lar 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 ap
plied in any given angle, and that length be as any power of the base
A^ ; and there be sought the force with which a body, either attracted to
wards the base or driven from it in the direction of that ordinate, may be
caused to move in the curre line which that ordinate always describes with
its superior extremity ;
I suppose the base to be increased by a very small
,m m
part O, and I resolve the ordinate A -f Ol^ into an infinite series A- -f
!!L OA^+ ^-^--- OOA ;- &c., and I suppose the force proper-
11 1111
tional to the term of this series in which O is of two dimensions, that is,
to the term - - OOA ^YT, Therefore the force sought is aa
SEC. XIV.J OF NATURAL PHILOSOPHY. 2M
mm mn m 2n .... mm mn m 2n A 7, , or, which is the same thinor, as L> m .
nn nn
As if the ordinate describe a parabola, m being 2, and n = 1, the force
will be as the given quantity 2B, and therefore is given. Therefore with
a given force the body will move in a parabola, as Galileo has demon
strated. If the ordinate describe an hyperbola, m being = 1, and n
1, the force will be as 2A 3 or 2B 3
; 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 fiot yet touched upon.
SECTION XIV.
Of the motion of very small bodies when agitated by centripetal forces
tending to the several parts of any very great body.
PROPOSITION XCIV. THEOREM XLVIII.
If two similar mediums be separatedfrom each other by a space termi
nated on both sides by parallel planes, and a body in its passage
through that space be attracted or impelled perpendicularly towards
either of those mediums, and not agitated or hindered by any other
force ; and the attraction be every where the same at equal distances
from either plane, taken towards the same hand of the plane ; I say,
that the sine of incidence upon either plane will be to the sine of emcr
gencefrom the other plane in a given ratio.
CASE 1. Let Aa and B6 be two parallel planes,
and let the body light upon the first plane Aa in
the direction of the line GH, and in its whole
passage through the intermediate space let it be
attracted or impelled towards the medium of in
cidence, and by that action let it be made to de
scribe a curve line HI, and let it emerge in the di
rection of the line IK. Let there be erected IM
perpendicular to Eb the plane of emergence, and
meeting the line of incidence GH prolonged in M, and the plane of inci
dence Aa in R ; and let the line of emergence KI be produced and meet
HM in L. About the centre L, with the interval LI, let a circle be de
scribed cutting both HM in P and Q, and MI produced in N ; and, first,
if the attraction or impulse be supposed uniform, the curve HI (by what
Galileo has demonstrated) be a parabola, whose property is that of a roc
THE MATHEMATICAL PRINCIPLES [BoOK 1
tangle under its given latiis rectum and the line IM is equal to the squarrf
cf HM ; and moreover the line HM will be bisected in L. Whence if to
MI there be let fall the perpendicular LO, MO, OR will be equal; and
adding the equal lines ON, OI, the wholes MN, IR will be equal also.
Therefore since IR is given, MN is also given, and the rectangle NMI is
to the rectangle under the latus rectum and IM, that is, to HMa in a given
ratio. But the rectangle NMI is equal to the rectangle PMQ,, that is, to
the difference of the squares ML2
, and PL2 or LI2
; and HM2 hath a given
ratio to its fourth part ML2
; therefore the ratio of ML2 LI2 to ML2
is given,
and by conversion the ratio of LI2 to ML , and its subduplicate, theratrio
of LI to ML. But in every triangle, as LMI, the sines jf the angles are
proportional to the opposite sides. Therefore the ratio of the sine of the
angle of incidence LMR to the sine of the angle of emergence LIR is
given. QJE.lr).
CASE 2. Let now the body pass successively through several spaces ter
minated with parallel planes Aa/>B, B6cC, &c., and let it be acted on by a
\ . force which is uniform in each of them separ-
\ a ately, but different in the different spaces ; and
B \ fr by what was just demonstrated, the sine of the
c ^^ c angle of incidence on the first plane Aa is to
the sine of emergence from the second plane Bb
in a given ratio
; and this sine of incidence upon the second plane Bb will
be to the sine of emergence from the third plane Cc in a given ratio
; and
this sine to the sine of emergence from the fourth plane Dd in a given ra
tio
; and so on in infinitum ; and, by equality, the sine of incidence on
the first plane to the sine of emergence from the last plane in a given ratio.
I ,et now the intervals of the planes be diminished, and their number be in
finitely increased, so that the action of attraction or impulse, exerted accord
ing to any assigned law, may become continual, and the ratio of the sine of
incidence on the first plane to the sine of emergence from the last plane
being all along given, will be given then also. QJE.D.
PROPOSITION XCV. THEOREM XLIX.
The same things being supposed, I say, that the velocity of the body be
fore its incidence is to its velocity after emergence as the sine of emer
gence to the sine of incid nee.
Make AH and Id equal, and erect the perpendiculars
AG, dK meeting the lines of incidence
and emergence GH, IK, in G and K. In GH
-- take TH equal to IK, and to the plane Aa let
^ fall a perpendicular TV. And (by Cor. 2 of the
|x^ I
Laws of Motion) let the motion of the body be
j v
- resolved into two, one perpendicular to the planes
SEC. X1V.J OF NATURAL PHILOSOPHY. 245
Aa, Bb, Cc, &c, and another parallel to them. The force of attraction or
impulse, acting in directions perpendicular to those planes, does not at all
alter the motion in parallel directions ; and therefore the body proceeding
with this motion will in equal times go through those equal parallel inter
vals that lie between the line AG and the point H, and between the point
I and the line dK ;
that is, they will describe the lines GH, IK in equal
times. Therefore the velocity before incidence is to the velocity after
emergence as GH to IK or TH, that is, as AH or Id to vH, that is (sup
posing TH or IK radius), as the sine of emergence to the sine of inci
dence. Q.E.D.
PROPOSITION XCVL THEOREM L.
The same things being supposed, and that the motion before incidence is
swifter than afterwards ; 1 sat/, lhat if the line of incidence be in
clined continually, the body will be at last reflected, and the angle of
reflexion will be equal to the angle of incidence.
For conceive the body passing between the parallel planes Aa, Bb, Cc,
&c., to describe parabolic arcs as above;
and let those arcs be HP, PQ, QR, &c.
And let the obliquity of the line of inci- g
dence GH to the first plane Aa be such rc~
that the sine of incidence may be to the radius of the circle whose sine it is,
in the same ratio which the same sine of incidence hath to the sine of emer
gence from the plane Dd into the space DefeE ; and because the sine of
emergence is now become equal to radius, the angle of emergence will be a
right one, and therefore the line of emergence will coincide with the plane
Dd. Let the body come to this plane in the point R ; and because the
line of emergence coincides with that plane, it is manifest that the body can
proceed no farther towards the plane Ee. But neither can it proceed in the
line of emergence Rd; because it is perpetually attracted or impelled towards
the medium of incidence. It will return, therefore, between the planes Cc,
Dd, describing an arc of a parabola Q,R</, whose principal vertex (by what
Galileo has demonstrated) is in R, cutting the plane Or in the same angle
at q, that it did before at Q, ; then going on in the parabolic arcs qp, ph,
&c., similar and equal to the former arcs QP, PH, &c., it will cut the rest
of the planes in the same angles at p, h, (fee., as it did before in P, H, (fee.,
and will emerge at last with the same obliquity at h with which it first
impinged on that plane at H. Conceive now the intervals of the planes
Aa, Bb, Cc, Dd, Ee, (fee., to be infinitely diminished, and the number in
finitely increased, so that the action of attraction or impulse, exerted ac
cording to any assigned law, may become continual; and, the angle of
emergence remaining all alor g equal to the angle of incidence, will be
equal to the same also at last. Q.E.D.
246 THE MATHEMATICAL PRINCIPLES IBoOK 1
SCHOLIUM.
These attractions bear a great resemblance to the reflexions and refrac
tions of light made in a given ratio of the secants, as was discovered h}
Siiellius ; and consequently in a given ratio of the sines, as was exhibited
by Hes Cortes. For it is now certain from the phenomena of Jupiter s
^satellites, confirmed by the observations of different astronomers, that light
is propagated in succession, and requires about seven or eight minutes to
travel from the sun to the earth. Moreover, the rays of light that are in
our air (as lately was discovered by Grimaldus, by the admission of light
into a dark room through a small hole, which 1 have also tried) in their
passage near the angles of bodies, whether transparent or opaque (such aa
the circular and rectangular edges of gold, silver and brass coins, or of
knives, or broken pieces of stone or glass), are bent or inflected round those
bodies as if they were attracted to them ; and those rays which in their
passage come nearest to the bodies are the most inflected, as if they were
most attracted : which tiling I myself have also carefully observed. And
those which pass at greater distances are less inflected
; and those at still
greater distances are a little inflected the contrary way, and form three
fringes of colours. In the figure 5 represents the edge of a knife, or any
-f:::r; N ^c :.-/ >V V U J
W~"~a~" "~a C: O la
kind of wedge AsB : and gowog,fmnif,emtme, dlsld, are rays inflected to
wards the knife in the arcs owo, nvn, mtm, Isl ; which inflection is greater
or less according; to their distance from the knife. Now since this inflec
tion of the rays is performed in the air without the knife, it follows that the
rays which fall upon the knife are first inflected in the air before they touch
the knife. And the case is the same of the rays falling upon glass. The
refraction, therefore, is made not in the point of incidence, but gradually, by
a continual inflection of the rays ; which is done partly in the air before they
touch the glass, partly (if [ mistake not) within the glass, after they have
entered it
;
as is represented in the rays ckzc, bujb^ ahxa, falling upon r,
q, p, and inflected between k and z, i and y, h and x. Therefore because
of the analogy there is between the propagation of the rays f light and the
motion of bodies, I thought it not amiss to add the followi g Propositions
far optical uses ; not at all. considering the nature of the rays of .light, or
inquiring whether they are bodies or not ; but only determining the tra
jectories of bodies which are extremely like the trajectories of the rays.
SEC. XIV.] OF NATURAL PHILOSOPHY. 247
PROPOSITION XCVII. PROBT.-EM XLVII.
Supposing t/w sine of incidence upon any superficies to be in a given ra
tio to the sine of emergence ; and that tha inflection of t/ts paths of
those bodies near that superficies is performed in a very short space,
which may be considered as a point ; it is required to determine suck
a superficies as may cause all the corpuscles issuing from any one
given place to converge to another given place.
Let A be the place from whence the cor- E
puscles diverge ; B the place to which they
should converge ; CDE the curve line which
by its revolution round the axis AB describes . /C
the superficies sought ; D, E, any two points of that curve ; and EF, EG,
perpendiculars let fall on the paths of the bodies AD, DB. Let the point
D approach to and coalesce with the point E ; and the ultimate ratio of
the line DF by which AD is increased, to the line DG by which DB is
diminished, will be the same as that of the sine of incidence to the sine of
emergence Therefore the ratio of the increment of the line AD to the
decrement of the line DB is given: and therefore if in the axis AB there
be taken any where the point C through which the curve CDE must
pass, and CM the increment of AC be taken in that given ratio to CN
the decrement of BC, and from the centres A, B, with the intervals AM,
BN, there be described two circles cutting each other in D ; that point D
will touch the curve sought CDE, and, by touching it any where at pleasure,
will determine that curve. Q.E.I.
COR. 1. By causing the point A or B to go off sometimes in infinitum,
and sometimes to move towards other parts of the point C, will bo obtain
ed all those figures which Cartesins has exhibited in his Optics and Geom
etry relating to refractions. The invention of which Cartcsius having
thought fit to conceal, is here laid open in this Proposition.
COR. 2. If a body lighting on any superfi
cies CD in the direction of a riO^ht line AD, Qj- \
drawn according to any law, should emerge
in the direction of another right line DK ;
and from the point C there be drawn curve
lines CP, CQ, always perpendicular to AD, DK ; the increments of the
lines PD, QD, and therefore the lines themselves PD, Q.D, generated by
those increments, will be as the sines of incidence and emergence to each
other, and e contra.
PROPOSITION XCVIII. PROBLEM XLVIII.
The same things supposed ; if round the axis AB any attractive super
ficies be described as CD, regular or irregular, through which the bo
dies issuing from the given place A must pass ; it is required to find
THE MATHEMATICAL PRINCIPLES. [BOOK J
a second attractive superficies EF, which may make those bodies con
verge to a given place B.
Let a line joining AB cut
the first superficies in C and
the second in E, the point D
being taken any how at plea
sure. And supposing the
f sine of incidence on the first
superficies to the sine of
emergence from the same, and the sine of emergence from the second super
ficies to the sine of incidence on the same, to be as any given quantity M
to another given quantity N; then produce AB to G, so that BG may be
to CE as M N to N ; and AD to H, so that AH may be equal to AG ;
arid DF to K, so that DK may be to DH as N to M. Join KB, and about
the centre D with the interval DH describe a circle meeting KB produced
in L, and draw BF parallel to DL; and the point F will touch the line
EF, which, being turned round the axis AB, will describe the superficies
sought. Q.E.F.
For conceive the lines CP, CQ to be every where perpendicular to AD,
