SPy13, are severally to the sev eral areas CSD, CBED,
SDEB, in the given ratio of the heights CP, CD, and
the area SP/B is proportional to the time in which
the body P will move through the arc P/B. the area
SDEB will be also proportional to that time. Let
the latus rectum of the hyperbola RPB be diminished
in infitiitum, the latus transversum remaining the
same; and the arc PB will come to coincide with the
right line CB, and the focus S, with the vertex B, Aand
the right line SD with the right line BD. And therefore the area
BDEB will be proportional to the time in which the body C, by its per
pendicular descent, describes the line CB. Q.E.I.
CASE 3. And by the like argument, if the figure
RPB is a parabola, and to the same principal ver
tex B another parabola BED is described, that
may always remain given while the former para
bola in whose perimeter the body P moves, by
having its latus rectum diminished and reduced
to nothing, comes to coincide with the line CB,
the parabolic segment BDEB will be proportional if
to the time in which that body P or C will descend to the centre S or B
Q.K.T
fl.l OF NATURAL PHILOSOPHY.
PROPOSITION XXXIII. THEOREM IX.
The tilings above found being supposed. I say, thai the, velocity of a Jai
ling body in any place C is to the velocity of a body, describing a
circle about the centre B at the distance BC, in, the subduplicate ratio
of AC, the distance of the body from the remoter vertex A of the circle
or rectangular hyperbola, to iAB, the principal semi-diameter of the
Let AB, the common dia
meter of both figures RPB,
DEB, be bisected in O; and
draw the right line PT that
may touch the figure RPB
in P, and likewise cut that
common diameter AB (pro
duced, if need be) in T; and
let SY be perpendicular to
this line, and BQ to this di
ameter, and suppose the latus
rectum of the figure RPB to
be L. Prom Cor. 9, Prop.
XVI, it is manifest that the
velocity of a body, moving
in the line RPB about the
centre S, in any place P, is
to the velocity of a body describing a circle about the same centre, at the
distance SP, in the subduplicate ratio of the rectangle L X SP to SY2
Por by the properties of the conic sections ACB is to CP2 as 2AO to L.
2CP5 X AO
and therefore rrrr; is equal to L. Therefore those, velocities an
o--
ACB
to each other in the subduplicate ratio of
CP3 X AO X SP
ACB toSY~. More
over, by the properties of the conic sections, CO is to BO as BO to TV.?
and (by composition or division) as CB to BT. Whence (by division cs
composition) BO or + CO will be to BO as CT to BT, that is, AC
CP2 X AO X SP
ACB"
will be to AO as CP to BQ; and therefore is equal to
~AO X BC * ^ W suPPose GV, tne breadth of the figure RPB, to
be diminished in infinitum, so as the point P may come to coincide with
the point C, and the point S with the point B. and the line SP with the
line BC, and the line SY with the line BQ; and the velocity of the body
now descending perpendicularly in the line CB will be to the velocity of
11
162 THE MATHEMATICAL PRINCIPLES [BOOK 1
a body describing a circle about the centre B, at the distance BC, in thr
BQ2 X AC X SP
subduplicate ratio of--r-^-^- to SY2
, that is (neglecting the ra- X Jo
tios of equality of SP to BC, and BQ,2 to SY2
), in the subduplicate ratio
of AC to AO, or iAB. Q.E.D.
COR. 1 . When the points B and S come to coincide, TC will become to
TS as AC to AO.
COR. 2. A body revolving in any circle at a given distance from the
centre, by its motion converted upwards, will ascend to double its distance
from the centre.
PROPOSITION XXXIV. THEOREM X.
If the. figure BED is a parabola, I say, that the velocity of a falling
body in any place C is equal to the velocity by which a body may
uniformly describe a circle about the centre B at half the interval BC
For (by Cor. 7, Prop. XVI) the velocity of a
body describing a parabola RPB about the cen
tre S, in any place P, is equal to the velocity of
a body uniformly describing a circle about the c
same centre S at half the interval SP. Let the
breadth CP of the parabola be diminished in
itifiiiitirni, so as the parabolic arc P/B may come
to coincide with the right line CB, the centre S s
with the vertex B, and the interval SP with the
interval BC, and the proposition will be manifest. Q.E.D.
PROPOSITION XXXV. THEOREM XL
The same things supposed, I say, that the area of the figure DES, de
scribed by the indefinite radius SD, is equal to the area which a body
with a radius equal to h df the latus rectum of the figure DES, by
uniformly revolving about the centre S, may describe in the same tijiw.
1 JD/
AJ
SEC. ni: OF NATURAL PHILOSOPHY.
For suppose a body C in the smallest moment of time describes in fal
ling the infinitely little line Cc. while another body K, uniformly revolv
ing about the centre S in the circle OK/r, describes the arc KA:. Erect the
perpendiculars CD, cd, meeting the figure DES in D, d. Join SD, Sf/.
SK. SA*; and draw Del meeting the axis AS in T, and thereon let fall the
perpendicular SY.
CASE 1. If the figure DES is a circle, or a rectangular hyperbola, bisect
its transverse diameter AS in O, and SO will be half the latus rectum.
And because TC is to TD as Cc to Dd, and TD to TS as CD to SY ;
ex aquo TC will be to TS as CD X Cc to SY X Dd. But (by Cor. 1,
Prop. XXXIII) TC is to TS as AC to AO; to wit, if in the coalescence
of the points D, d, the ultimate ratios of the lines are taken. Wherefore
AC is to AO or SK as CD X Cc to S Y X Vd, Farther, the velocity of
the descending body in C IF, to the velocity of a body describing a circle
about the centre S, at the interval SC, in the subduplicate ratio of AC to
AO or SK (by Pi-op. XXXIII) ; and this velocity is to the velocity of a
body describing the circle OKA: in the subduplicate ratio of SK to SC
(by Cor. 6, Prop IV) ; and, ex aqnnj the first velocity to the last, that is,
the little line Cc to the arc K/r, in the subduplicate ratio of AC to SC,
that is, in the ratio of AC to CD. Wherefore CD X Cc is equal to AC
X KA*, and consequently AC to SK as AC X KA: to SY X IW. and
thence SK X KA: equal to SY X Drf, and iSK X KA: equal to SY X DC/,
that is, the area KSA* equal to the area SDrf. Therefore in every moment
of time two equal particles, KSA" and SDrf, of areas are generated, which,
if their magnitude is diminished, and their number increased in iiifinif t-w,
obtain the ratio cf equality, and consequently (by Cor. Lem. IV), the whole
areas together generated are always equal. Q..E.D.
CASE 2. But if the figure DES is a
parabola, we shall find, as above. CD X
Cc to SY X Df/ as TC to TS, that is,
as 2 to 1
; and that therefore |CD X Cc
is equal to i SY X Vd. But the veloc
ity of the falling body in C is equal to
the velocity writh which a circle may be
uniformly described at the interval 4SC
(by Prop" XXXIV). And this velocity
to the velocity with which a circle may
be described with the radius SK, that is,
the little line Cc to the arc KA,
is (by
Cor. 6, Prop. IV) in the subduplicate ratio of SK to iSC ; that is, in the
ratio of SK to *CD. Wherefore iSK X KA: is equal to 4CD X Cc, and
therefore equal to SY X T)d ; that is, the area KSA* is equal to the area
SIW, as above. Q.E.D.
164 THE MATHEMATICAL PRINCIPLES [BOOK 1.
PROPOSITION XXXVI. PROBLEM XXV.
To determine the times of the descent of a body falling from
place A.
Upon the diameter AS, the distance of the body from the
centre at the beginning, describe the semi-circle ADS, as
likewise the semi-circle OKH equal thereto, about the centre
S. From any place C of the body erect the ordinate CD.
Join SD, and make the sector OSK equal to the area ASD.
It is evident (by Prop. XXXV) that the body in falling will
describe the space AC in the same time in which another body,
uniformly revolving about the centre S, may describe the arc
OK. Q.E.F. M
a given
PROPOSITION XXXVII. PROBLEM XXVI.
To define the times of the ascent or descent of a body projected upwards
or downwards from a given place.
Suppose the body to go oif from the given place G, in the direction of
the line GS, with any velocity. In the duplicate ratio of this velocity to
the uniform velocity in a circle, with which the body may revolve about
\
H
D
the centre S at the given interval SG, take GA to AS. If that ratio is
the same as of the number 2 to 1, the point A is infinitely remote ; in
which case a parabola is to be described with any latus rectum to the ver
tex S, and axis SG ; as appears by Prop. XXXIV. But if that ratio is
less or greater than the ratio of 2 to 1, in the former case a circle, in the
latter a rectangular hyperbola, is to be described on the diameter SA; as
appears by Prop. XXXIII. Then about the centre S, with an interval
equal to half the latus rectum, describe the circle H/vK ; and at the place
G of the ascending or descending body, and at any other place C, erect the
perpendiculars GI, CD, meeting the conic section or circle in I and D.
Then joining SI, SD, let the sectors HSK, HS& be made equal to the
segments SEIS, SEDS. and (by Prop. XXXV) the body G will describe
SEC. VII.] OF NATURAL PHILOSOPHY. 165
the space GO in the same time in which the body K may describe t*he arc
Kk. Q.E.F.
PROPOSITION XXXVIII. THEOREM XII.
Supposing that the centripetal force is proportional to the altitude or
distance ofplaces from the centre, I say, that the times and velocities
offalling bodies, and the spaces which they describe, are respectively
proportional to the arcs, and the right and versed sines of the arcs.
Suppose the body to fall from any place A in the A.
right line AS ; and about the centre of force S, with
the interval AS, describe the quadrant of a circle AE ;
and let CD be the right sine of any arc AD ; and the
body A will in the time AD in falling describe the
space AC, and in the place C will acquire the ve
locity CD.
This is demonstrated the same way from Prop. X, as Prop. XXX11 was
demonstrated from Prop. XI.
COR. 1. Hence the times are equal in which one body falling from the
place A arrives at the centre S, and another body revolving describes the
quadrantal arc ADE.
COR. 2. Wherefore all the times are equal in which bodies falling from
whatsoever places arrive at the centre. For all the periodic times of re
volving bodies are equal (by Cor. 3; Prop. IV).
PROPOSITION XXXIX. PROBLEM XXVIT.
Supposing a centripetal force of any kind, and granting the quadratnres
of curvilinear figures ; it is required to find the velocity of a bod)/,
ascending or descending in a right line, in the several places through
which it passes ; as also the time in which it will arrive at any place :
and vice versa.
Suppose the body E to fall from any place
A in the right line ADEC ; and from its place
E imagine a perpendicular EG always erected
proportional to the centripetal force in that
place tending to the centre C ; and let BFG
be a curve line, the locus of the point G. And D
in the beginning of the motion suppose EG to
coincide with the perpendicular AB ; and the
velocity of the body in any place E will be as
a right line whose square is equal to the cur
vilinear area ABGE. Q.E.I.
In EG take EM reciprocally proportional to
E
366 THE MATHEMATICAL PRINCIPLES [BOOK 1
a right line whose square is equal to the area ABGE, and let VLM be a
curve line wherein the point M is always placed, and to which the right
line AB produced is an asymptote; and the time in which the body in
falling- describes the line AE, will be as the curvilinear area ABTVME.
Q.E.I.
For in the right line AE let there be taken the very small line DE of
a given length, and let DLF be the place of the line EMG, when the
body was in D ; and if the centripetal force be such, that a right line,
whose square is equal to the area ABGE; is as the velocity of the descend
ing body, the area itself will be as the square of that velocity ; that is, if
for the velocities in D and E we write V and V + I, the area ABFD will
be as VY, and the area ABGE as YY + 2VI -f II; and by division, the
area DFGE as 2VI -f LI, and therefore ^ will be as--^r
that is. if we take the first ratios of those quantities when just nascent, the
2YI
length DF is as the quantity -|yrr
an(i therefore also as half that quantity
1 X Y
But the time in which the body in falling describes the very
line DE, is as that line directly and the velocity Y inversely ; and
the force will be as the increment I of the velocity directly and the time
inversely ; and therefore if we take the first ratios when those quantities
I X V
are just nascent, as
-jy==r-.
that is, as the length DF. Therefore a force
proportional to DF or EG will cause the body to descend with a velocity
that is as the right line whose square is equal to the area ABGE. Q.E.D.
Moreover, since the time in which a very small line DE of a given
length may be described is as the velocity inversely, and therefore also
inversely as a right line whose square is equal to the area ABFD ; and
since the line DL. and by consequence the nascent area DLME, will be as
(he same right line inversely, the time will be as the area DLME, and
the sum of all the times will be as the sum of all the areas : that is (by
Cor. Lern. IV), the whole time in which the line AE is described will be
as the whole area ATYME. Q.E.D.
COR. 1. Let P be the place from whence a body ought to fall, so as
that, when urged by any known uniform centripetal force (such as
gravity is vulgarly supposed to be), it may acquire in the place D a
velocity equal to the velocity which another body, falling by any force
whatever, hath acquired in that place D. In the perpendicular DF let
there be taken DR., which may be o DF as that uniform force to
the other force in the place D. Complete the rectangle PDRQ,, and cut
iff the area. ABFD equal to that rectangle. Then A will be the place
SEC. VII. I OF NATURAL PHILOSOPHY. 10;
from whence the other body fell. For com
pleting the rectangle DRSE, since the area
ABFD is to the area DFGE as VV to 2VI,
and therefore as 4V to I, that is, as half the
whole velocity to the increment of the velocity
of the body falling by the unequable force
; and
in like manner the area PQRD to the area
DRSE as half the whole velocity to the incre
ment of the velocity of the body falling by the
uniform force ; and since those increments (by
reason of the equality of the nascent times)
are as the generating forces, that is, as the ordinates
DF, DR, and consequently as the nascent areas DFGE, DRSE :
therefore, ex aq-uo, the whole areas ABFD, PQRD will be to one another
as the halves of the whole velocities ; and therefore, because the velocities
are equal, they become equal also.
COR. 2. Whence if any body be projected either upwards or downwards
with a given velocity from any place D, and there be given the law of
centripetal force acting on it, its velocity will be found in any other place,
as e, by erecting the ordinate eg, and taking that velocity to the velocity
in the place D as a right line whose square is equal to the rectangle
PQRD, either increased by the curvilinear area DFge, if the place e is
below the place D, or diminished by the same area DFg-e, if it be higher,
is to the right line whose square is equal to the rectangle PQRD alone.
COR. 3. The time is also known by erecting the ordinate em recipro
cally proportional to the square root of PQRD -f- or T)Fge, and taking
the time in which the body has described the line De to the time in which
another body has fallen with an uniform force from P, and in falling ar
rived at D in the proportion of the curvilinear area DLme to the rectan
gle 2PD X DL. For the time in which a body falling with an uniform
force hath described the line PD, is to the time in which the same body
has described the line PE in the subduplicate ratio of PD to PE ; that is
(the very small line DE being just nascent), in the ratio of PD to PD -f
^DE; or 2PD to 2PD -f- DE, and, by division, to the time in which the
body hath described the small line DE, as 2PD to DE, and therefore as
the rectangle 2PD X DL to the area DLME ; and the time in which
both the bodies described the very small line DE is to the time in which
the body moving unequably hath described the line De as the area DLME
to the area DLme ; and, ex aquo, the first mentioned of these times is to
the last as the rectangle 2PD X DL to the area DLrae.
163 THE MATHEMATICAL PRINCIPLES [BoOK I
SECTION VIII.
Of the invention of orbits wherein bodies will revolve, being acted upon
by any sort of centripetal force.
PROPOSITION XL. THEOREM XIII.
// a body, acted upon by any centripetal force, is any how moved, and
another body ascends or descends in a right line, and their velocities
be equal in amj one case of equal altitudes, t/ieir velocities will be also
equal at all equal altitudes.
Let a body descend from A through D and E, to the centre
(j : and let another body move from V in the curve line VIK&.
From the centre C, with any distances, describe the concentric
circles DI, EK, meeting the right line AC in I) and E; and
the curve VIK in I and K. Draw 1C meeting KE in N, and
on IK let fall the perpendicular NT and let the interval DE
or IN between the circumferences of the circles be very small ;
K
/
and imagine the bodies in D and I to have equal velocities.
Then because the distances CD and CI are equal, the centri
petal forces in D and I will be also equal. Let those forces be k)
expressed by the equal lineoke DE and IN ; and let the force
IN (by Cor. 2 of the Laws of Motion) be resolved into two
others, NT and IT. r
l hen the force NT acting in the direction
line NT perpendicular to the path ITK of the body will not at all affect
or change the velocity of the body in that path, but only draw it aside
from a rectilinear course, and make it deflect perpetually from the tangent
of the orbit, and proceed in the curvilinear path ITK/j. That whole
force, therefore, will be spent in producing this effect: but the other force
IT, acting in the direction of the course of the body, will be all employed
in accelerating it, and in the least given time will produce an acceleration
proportional to itself. Therefore the accelerations of the bodies in D and
I, produced in equal times, are as the lines DE, IT (if we take the first
ratios of the nascent lines DE, IN, IK, IT, NT) ; and in unequal times as
those lines and the times conjunctly. But the times in which DE and IK
are described, are, by reason of the equal velocities (in D and I) as the
spaces described DE and IK, and therefore the accelerations in the course
of the bodies through the lines DE and IK are as DE and IT, and DE
and IK conjunctly ; that is, as the square of DE to the rectangle IT into
IK. But the rectangle IT X IK is equal to the square of IN, that is,
equal to the square of DE ; and therefore the accelerations generated in
the passage of the bodies from D and I to E and K are equal. Therefore
the velocities of the holies in E and K are also equal, and by the same
reasoning they will always be found equal in any subsequent equal dis
tances. Q..E.D.
SEC. VI11.J OF NATURAL PHILOSOPHY. 169
By the same reasoning, bodies of equal velocities and equal distances
from the centre will be equally retarded in their ascent to equal distances.
Q.E.D.
COR. 1. Therefore if a body either oscillates by hanging to a string, or
by any polished and perfectly smooth impediment is forced to move in a
curve line ; and another body ascends or descends in a right line, and their
velocities be equal at any one equal altitude, their velocities will be also
equal at all other equal altitudes. For by the string of the pendulous
body, or by the impediment of a vessel perfectly smooth, the same thing
will be effected as by the transverse force NT. The body is neither
accelerated nor retarded by it, but only is obliged to leave its rectilinear
course.
COR. 2. Suppose the quantity P to be the greatest distance from the
centre to which a body can ascend, whether it be oscillating, or revolving
in a trajectory, and so the same projected upwards from any point of a
trajectory with the velocity it has in that point. Let the quantity A be
the distance of the body from the centre in any other point of the orbit ; and
let the centripetal force be always as the power An
, of the quantity A, the
index of which power n 1 is any number n diminished by unity. Then
the velocity in every altitude A will be as v/ P11
A", and therefore will
be given. For by Prop. XXXIX, the velocity of a body ascending and
descending in a right line is in that very ratio.
PROPOSITION XLI. PROBLEM XXVTII.
Supposing a centripetal force of any kind, and granting the quadra
tures of curvilinear figures, it is required to find as well the trajecto
ries in which bodies will move, as the times of their motions in the
trajectories found.
Let any centripetal force tend to
the centre C, and let it be required
to find the trajectory VIKAr. Let R,
