same be bisected in S, the lenc th of the part PS will be to the length PV
(which is the double of the sine of the angle YBP, when EB is radius) as
2CE to CB, and therefore in a given ratio.
COR. 2. And the length of the semi-perimeter of the cycloid AS will be
equal to a right line which is to the dumeter of the wheel BY as 2CF
toCB.
PROPOSITION L. PROBLEM XXXIII.
To cause a pendulous body to oscillate in a given cycloid.
Let there be given within the globe QYS described
with the centre C, the cycloid QRS, bi
sected in R, and meeting the superficies of the
globe with its extreme points Q and S on either
hand. Let there be drawn CR birxcting the arc
QS in O, and let it be produced to A in such
sort that CA may be to CO as CO to CR.
About the centre C, with the interval CA, let
there be described an exterior globe DAF ; and
within this globe, by a wheel whose diameter is
AO, let there be described two semi-cycloids AQ,,
AS, touching the interior globe in Q, and S, and meeting the exterior globe
in A. From that point A, with a thread APT in length equal to the line
AR, let the body T depend, and oscillate in such manner between the two
SlCC. X.J OF NATURAL PHILOSOPHY. 187
semi-cycloids AQ, AS, that, as often as the pendulum parts from the per
pendicular AR, the upper part of the thread AP may be applied to that
semi-cycloid APS towards which the motion tends, and fold itself round
that curve line, as if it were some solid obstacle, the remaining part of the
same thread PT which has not yet touched the semi-cycloid continuing
straight. Then will the weight T oscillate in the given cycloid QRS.
Q.E.F.
For let the thread PT meet the cycloid QRS in T, and the circle QOS
m V, and let 0V be drawn j and to the rectilinear part of the thread PT
from the extreme points P and T let there be erected the perpendiculars
BP, TW, meeting the right line CV in B and W. It is evident, from the
construction and generation of the similar figures AS, SR, that those per
pendiculars PB, TVV, cut off from CV the lengths VB, VVV equal the
diameters of the wheels OA, OR. Therefore TP is to VP (which is dou
ble the sine of the angle VBP when ^BV is radius) as BYV to BV, or AO
-f-OR to AO, that is (since CA and CO, CO and CR; and by division AO
and OR are proportional), as CA + CO to CA, or, if BV be bisected in E,
as 2CE to CB. Therefore (by Cor. 1, Prop. XLIX), the length of the
rectilinear part of the thread PT is always equal to the arc of the cycloid
PS, and the whole thread APT is always equal to the half of the cycloid
APS, that is (by Cor. 2, Prop. XLIX), to the length AR. And there
fore contrariwise, if the string remain always equal to the length AR, the
point T will always move in the given cycloid QRS. Q.E.D.
COR. The string AR is equal to the semi-cycloid AS, and therefore has
the same ratio to AC the semi-diameter of the exterior globe as the like
semi-cycloid SR has to CO the semi-diameter of the interior globe.
PROPOSITION LI. THEOREM XVIII.
If a centripetal force tending on all sides to the centre C of a globe, be in
all places as the distance of the place from the centre, and by thisforce
alone acting upon it, the body T oscillate (in the manner above de
scribed] in the perimeter of the cycloid QRS ; / say, that all the oscil
lations, how unequal soever in tfiemselves, will be performed in equal
times.
For upon the tangent TW infinitely produced let fall the perpendicular
CX, and join CT. Because the centripetal force with which the body T
is impelled towards C is as the distance CT, let this (by Cor. 2, of the
I ,aws) be resolved into the parts CX, TX, of which CX impelling the
body directly from P stretches the thread PT, and by the resistance the
rhread makes to it is totally employed, producing no other effect
; but the
3ther part TX, impelling the body transversely or towards X, directly
accelerates the motion in the cycloid. Then it is plain that the accelera
tion of the body, proportional to this accelerating force, will bo every
188 THE MATHEMATICAL PRINCIPLES [BOOK 1
moment as the length TX, that is (because CV\
WV, and TX, TW proportional to them are given),
as the length TW, that is (by Cor. 1, Prop. XLIX)
as the length of the arc of the cycloid TR. If there
fore two pendulums APT, Apt, be unequally drawn
aside from the perpendicular AR, and let fall together,
their accelerations will be always as the arcs to be de
scribed TR, tR. But the parts described at the
beginning of the motion are as the accelerations, thai
is, as the wholes that are to be described at the be
ginning, and therefore the parts which remain to be
described, and the subsequent accelerations proportional to those parts, are
also as the wholes, and so on. Therefore the accelerations, and consequently
the velocities generated, and the parts described with those velocities, and
the parts to be described, are always as the wholes ; and therefore the parts
to be described preserving a given ratio to each other will vanish together,
that is, the two bodies oscillating will arrive together at the perpendicular AR.
And since on the other hand the ascent of the pendulums from the lowest place
R through the same cycloidal arcs with a retrograde motion, is retarded in
the several places they pass through by the same forces by which their de
scent was accelerated : it is plain that the velocities of their ascent and de
scent through the same arcs are equal, and consequently performed in equal
times ; and, therefore, since the two parts of the cycloid RS and RQ lying
on either side of the perpendicular are similar and equal, the two pendu
lums will perform as well the wholes as the halves of their oscillations in
the same times. Q.E.D.
COR. The force with which the body T is accelerated or retarded in any
place T of the cycloid, is to the whole weight of the same body in the
highest place S or Q, as the arc of the cycloid TR is to the arc SR or QR
PROPOSITION LIL PROBLEM XXXIV.
To define the velocities of the pendulums in the several places, and the
times in which both the entire oscillations, and the several parts of them
are performed.
About any centre G, with the interval GH equal to
the arc of the cycloid RS, describe a semi-circle HKM
bisected by the semi-diameter GK. And if a centripe
tal force proportional to the distance of the places from
the centre tend to the centre G, and it be in the peri
meter HIK equal to the centripetal force in the perime
ter of the globe Q,OS tending towards its centre, and at
the same time that the pendulum T is let fall from the
highest place S, a body, as L, is let fall from H to G ; then because th<
SEC. X.J OF NATURAL PHILOSOPHY. 189
forces which act upon the bodies are equal at the be
ginning, and always proportional to the spaces to be
described TR, LG, and therefore if TR and LG are
equal, are also equal in the places T and L, it is plain
that those bodies describe at the beginning equal spaces
M
ST, HL, and therefore are still acted upon equally, and continue to describe
equal spaces. Therefore by Prop. XXXVIII, the time in which the body
describes the arc ST is to the time of one oscillation, as the arc HI the time
in which the body H arrives at L, to the semi-periphery HKM, the time
in which the body H will come to M. And the velocity of the pendulous
body in the place T is to its velocity in the lowest place R, that is, the
velocity of the body H in the place L to its velocity in the place G, or the
momentary increment of the line HL to the momentary increment of the
line HG (the arcs HI, HK increasing with an equable flux) as the ordinato
LI to the radius GK. or as v/SR2 Til2 to SR. Hence, since in unequal
oscillations there are described in equal time arcs proportional to the en
tire arcs of the oscillations, there are obtained from the times given, both
the velocities and the arcs described in all the oscillations universally.
Which was first required.
Let now any pendulous bodies oscillate in different cycloids described
within different globes, whose absolute forces are also different ; and if the
absolute force of any globe Q.OS be called V, the accelerative force with
which the pendulum is acted on in the circumference of this globe, when it
begins to move directly towards its centre, will be as the distance of the
pendulous body from that centre and the absolute force of the globe conjunctly,
that is, as CO X V. Therefore the lineola HY, which is as this
accelerated force CO X V, will be described in a given time : and if there
be erected the perpendicular YZ meeting the circumference in Z, the nascent
arc HZ will denote that given time. But that nascent arc HZ is in the
subduplicate ratio of the rectangle GHY, and therefore as v/GH X CO X V
Whence the time of an entire oscillation in the cycloid Q,RS (it being as
the semi-periphery HKM, wrhich denotes that entire oscillation, directly :
and as the arc HZ which in like manner denotes a given time inversely)
will be as GH directly and v/GH X CO X V inversely ; that is, because
GH and SR are equal, as VnUrU, . or (by Cor. Prop. L,) as X/-TTVT- X V AO X V
Therefore the oscillations in all globes and cycloids, performed with what
absolute forces soever, are in a ratio compounded of the subduplicate ratio of
the length of the string directly, and the subduplicate ratio of the distance
between the point of suspension and the centre of the globe inversely, and
the subduplicate ratio of the absolute force of the globe inversely also
Q.E.I.
t90 THE MATHEMATICAL PRINCIPLES [Bo^K 1.
COR. 1. Hence also the times of oscillating, falling, and revolving bodies
may be compared among themselves. For if the diameter of the wheel
with which the cycloid is described within the globe is supposed equal to
the semi-diameter of the globe, the cycloid will become a right line passing
through the centre of the globe, and the oscillation will be changed into a
descent and subsequent ascent in that right line. Whence there is given
both the time of the descent from any place to the centre, and the time equal
to it in which the body revolving uniformly about the centre of the globe
at any distance describes an arc of a quadrant For this time (by
Case 2) is to the time of half the oscillation in any cycloid QJR.S as 1 to
AR
V AC
COR. 2. Hence also follow what Sir Christopher Wren and M. Huygevs
have discovered concerning the vulgar cycloid. For if the diameter of the
globe be infinitely increased, its sphacrical superficies will be changed into a
plane, and the centripetal force will act uniformly in the direction of lines
perpendicular to that plane, and this cycloid of our s will become the same
with the common cycloid. But in that case the length of the arc of the
cycloid between that plane and the describing point will become equal to
four times the versed sine of half the arc of the wheel between the same
plane and the describing point, as was discovered by Sir Christopher Wren.
And a pendulum between two such cycloids will oscillate in a similar and
equal cycloid in equal times, as M. Huygens demonstrated. The descent
of heavy bodies also in the time of one oscillation will be the same as M.
Huygens exhibited.
The propositions here demonstrated are adapted to the true constitution
of the Earth, in so far as wheels moving in any of its great circles will de
scribe, by the motions of nails fixed in their perimeters, cycloids without the
globe ; and pendulums, in mines and deep caverns of the Earth, must oscil
late in cycloids within the globe, that those oscillations may be performed
in equal times. For gravity (as will be shewn in the third book) decreases
in its progress from the superficies of the Earth ; upwards in a duplicate
ratio of the distances from the centre of the Earth ; downwards in a sim
ple ratio of the same.
PROPOSITION LIII. PROBLEM XXXV.
Granting the quadratures of curvilinear figures, it is required to find
the forces with which bodies moving in given curve lines may always
perform their oscillations in equal times.
Let the body T oscillate in any curve line STRQ,, whose axis is AR
passing through the centre of force C. Draw TX touching that curve in
any place of the body T, and in that tangent TX take TY equal to the
arc TR. The length of that arc is known from the common methods used
SEC. X. OF NATURAL PHILOSOPHY. 191
for the quadratures of figures. From the point Y
draw the right line YZ perpendicular to the tangent.
Draw CT meeting that perpendicular in Z, and the
centripetal force will be proportional to the right line
TZ. Q.E.I.
For if the force with which the body is attracted
from T towards C be expressed by the right line TZ
taken proportional to it, that force will be resolved
into two forces TY, YZ, of which YZ drawing the
body in the direction of the length of the thread PT,
docs not at all change its motion ; whereas the other
force TY directly accelerates or retards its mction in the curve STRQ.
Wherefore since that force is as the space to be described TR, the acceler
ations or retardations of the body in describing two proportional parts (u
greater arid a less) of two oscillations, will be always as those parts, and
therefore will cause those parts to be described together. But bodies which
continually describe together parts proportional to the wholes, will describe
the wholes together also. Q,.E.l).
COR. 1. Hence if the body T, hanging by a rectilinear thread
AT from the centre A, describe the circular arc STRQ,,
and in the mean time be acted on by any force tending
downwards with parallel directions, which is to the uni
form force of gravity as the arc TR to its sine TN, the
times of the several oscillations will be equal. For because
TZ, AR are parallel, the triangles ATN, ZTY are similar ; and there
fore TZ will be to AT as TY to TN ; that is, if the uniform force of
gravity be expressed by the given length AT, the force TZ. by which the
oscillations become isochronous, will be to the force of gravity AT, as the
arc TR equal to TY is to TN the sine of that arc.
COR. 2. And therefore in clocks, if forces were impressed by some ma
chine upon the pendulum which preserves the motion, and so compounded
with the force of gravity that the whole force tending downwards should
be always as a line produced by applying the rectangle under the arc TR
and the radius AR to the sine TN, all the oscillations will become
isochronous.
PROPOSITION LIV. PROBLEM XXXYI.
Granting the quadratures of curvilinear figures, it is required to find
the times in which bodies by means of any centripetal force will descend
or ascend in any curve lines described in, a plane passing through the
centre of force.
Let the body descend from any place S, and move in any curve ST/R
given in a plane passing through the centre of force C. Join CS, and lei
192 THE MATHEMATICAL PRINCIPLES [BOOK 1
Q it be divided into innumerable equal parts, and let
Dd be one of those parts. From the centre C, with
the intervals CD, Cd, let the circles DT, dt be de
scribed, meeting the curve line ST*R in T and t.
And because the law of centripetal force is given.
and also the altitude CS from which the body at
first fell, there will be given the velocity of the body
in any other altitude CT (by Prop. XXXIX). But
the time in which the body describes the lineola Tt
is as the length of that lineola, that is, as the secant
of the angle /TC directly, and the velocity inversely.
Lei, the ordinate DN, proportional to this time, be made perpendicular to
the right line CS at the point D, and because Dd is given, the rectangle
Dd X DN, that is, the area DNwc?, will be proportional to the same time.
Therefore if PN/?, be a curve line in which the point N is perpetually found,
and its asymptote be the right line SQ, standing upon the line CS at right
angles, the area SQPJN D will be proportional to the time in which the body
in its descent hath described the line ST ; and therefore that area being
found, the time is also given. Q.E.I.
PROPOSITION LV. THEOREM XIX.
If a body move in any curve superficies, whose axis passes through the
centre offorce, and from the body a perpendicular be let fall iipon the
axis \ and a line parallel and equal thereto be drawn from any given
point of the axis ; I say, that this parallel line will describe an area
proportional to the time.
Let BKL be a curve superficies, T a body
revolving in it, STR a trajectory which the
body describes in the same, S the beginning
of the trajectory, OMK the axis of the curve
superficies, TN a right line let fall perpendic
ularly from the body to the axis ; OP a line
parallel and equal thereto drawn from the
given point O in the axis ; AP the orthogra
phic projection of the trajectory described by
the point P in the plane AOP in which the
revolving line OP is found : A the beginning
of that projection, answering to the point S ;
TO a right line drawn from the body to the centre ; TG a part thereof
proportional to the centripetal force with which the body tends towards the
centre C ; TM a right line perpendicular to the curve superficies ; TI a
part thereof proportional to the force of pressure with which the body urges
SEC. X.] OF NATURAL PHILOSOPHY. 193
the superficies, and therefore with which it is again repelled by the super
ficies towards M ; PTF a right line parallel to the axis and passing through
the body, and OF, IH right lines let fall perpendicularly from the points
G and I upon that parallel PHTF. I say, now. that the area AGP, de
scribed by the radius OP from the beginning of the motion, is proportional
to the time. For the force TG (by Cor. 2, of the Laws of Motion) is re
solved into the forces TF, FG ; and the force TI into the forces TH, HI ;
but the forces TF, TH, acting in the direction of the line PF perpendicular
to the plane AOP, introduce no change in the motion of the body but in a di
rection perpendicular to that plane. Therefore its motion, so far as it has
the same direction with the position of the plane, that is, the motion of the
point P, by which the projection AP of the trajectory is described in that
plane, is the same as if the forces TF, TH were taken away, and the body
wei e acted on by the forces FG, HI alone ; that is, the same as ,f the body
were to describe in the plane AOP the curve AP by means of a centripetal
force tending to the centre O, and equal to the sum of the forces FG and
HI. But with such a force as that (by Prop. 1) the area AOP will be de
scribed proportional to the time. Q.E.D.
COR. By the same reasoning, if a body, acted on by forces tending to
two or more centres in any the same right line CO, should describe in a
free space any curve line ST, the area AOP would be always proportional
to the time.
PROPOSITION LVI. PROBLEM XXXVII.
Granting the quadratures of curvilinear figures, and supposing that
there are given both the law of centripetal force tending to a given cen
tre, and the curve superficies whose axis passes through that centre
;
it is required to find the trajectory which a body will describe in that
superficies, when going offfrom a given place with a given velocity,
and in a given direction in that superficies.
The last construction remaining, let the
body T go from the given place S, in the di
rection of a line given by position, and turn
into the trajectory sought STR, whose ortho
graphic projection in the plane BDO is AP.
And from the given velocity of the body in
the altitude SC, its velocity in any other al
titude TC will be also given. With that
velocity, in a given moment of time, let the
body describe the particle Tt of its trajectory,
and let P/? be the projection of that particle
described in the plane AOP. Join Op, and
a little circle being described upon the curve superficies about the centre T
13
194 THE MATHEMATICAL PRINCIPLES [BOOK I
with the interval TV let the projection of that little circle in the plane AOP
be the ellipsis pQ. And because the magnitude of that little circle T/, and
TN or PO its distance from the axis CO is also given, the ellipsis pQ, will
be given both in kind and magnitude, as also its position to the right line
PO. And since the area PO/? is proportional to the time, and therefore
given because the time is given, the angle POp will be given. And thence
will be given jo the common intersection of the ellipsis and. the right line
Op, together with the angle OPp, in which the projection APp of the tra
jectory cuts the line OP. But from thence (by conferring Prop. XLI, with
Us 2d Cor.) the mariner of determining the curve APp easily appears.
Then from the several points P of that projection erecting to the plane
AOP, the perpendiculars PT meeting the curve superficies in T, there will
be iven the several points T of the trajectory. Q.E.I.
SECTION XL
f f the motions of bodies tending to each other with centripetal forces.
I have hitherto been treating of the attractions of bodies towards an im
movable centre; though very probably there is no such thing existent in
nature. For attractions are made towards bodies, and the actions of the
bodies attracted and attracting are always reciprocal and equal, by Law III
;
BO that if there are two bodies, neither the attracted nor the attracting body
is truly at rest, but both (by Cor. 4, of the Laws of Motion), being as it
were mutually attracted, revolve about a common centre of gravity. And
if there be more bodies, which are either attracted by one single one which
is attracted by them again, or which all of them, attract each other mutu
ally , these bodies will be so moved among themselves, as that their common
centre of gravity will either be at rest, or move uniformly forward in a
right line. I shall therefore at present go on to treat of the motion of
bodies mutually attracting each other ; considering the centripetal forces
as attractions ; though perhaps in a physical strictness they may more truly
be called impulses. But these propositions are to be considered as purely
mathematical; and therefore, laying aside all physical considerations, I
make use of a familiar way of speaking, to make myself the more easily
understood by a mathematical reader.
PROPOSITION LVII. THEOREM XX.
Two bodies attracting each other mutually describe similarfigures about
their common centre of gravity, and about each other mutually.
For the distances of the bodies from their common centre of gravity are
leciprocally as the bodies; and therefore in a given ratio to each other:
*nd thence, bv composition of ratios, in a given ratio to the whole distance
SEC. XL] OF NATURAL PHILOSOPHY. 195
between the bodies. Now these distances revolve about their common term
with an equable angular motion, because lying in the same right line they
never change their inclination to each other mutually But right lines
that are in a given ratio to each other, and revolve about their terms with
an equal angular motion, describe upon planes, which either rest with
those terms, or move with any motion not angular, figures entirely similar
round those terms. Therefore the figures described by the revolution ot
these distances are similar. Q.E.D.
PROPOSITION LVIll. THEOREM XXI.
If two bodies attract each other mutually with forces of any kind, and
in the mean time revolve about the common centre of gravity ; I say,
that, by the same forces, there may be described round either body un
moved ajigure similar and equal to the figures ivhich the bodies so
moving describe round each other mutually.
Let the bodies S and P revolve about their common centre of gravity
C, proceeding from S to T, and from P to Q,. From the given point s lot
there be continually drawn sp, sq, equal and parallel to SP, TQ, ; and the
;urve pqv, which the point p describes in its revolution round the immovable
point s, will be similar and equal to the curves which the bodies S and P
describe about each other mutually ; and therefore, by Theor. XX, similar
to the curves ST and PQ,V which the same bodies describe about their
common centre of gravity C and that because the proportions of the lines
SC. CP, and SP or sp, to each other, are given.
CASE 1. The common centre of gravity C (by Cor. 4, of the Laws of Mo
