Newton's Demonstration that planets move in ellipsesIsaac Newtonc.2,394 wordsThe Newton ProjectFalmer2010Newton Project, University of Sussex
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August 1684, in English, c. 2,574 words, 5 pp.5 pp.
in English
MS Add. 3965.1, ff. 1r-3v, Cambridge University Library, Cambridge, UKUKCambridgeCambridge University LibraryPortsmouth CollectionMS Add. 3965.1, ff. 1r-3vAugust 1684EnglandEnglishUnknown HandSciencePhysicsCatalogue information compiled by Rob Iliffe, Peter Spargo & John YoungTagged transcription by Yvonne SantacreuProofed by Robert IliffeTranscription checked by Michael HawkinsCatalogue exported to teiHeader by Michael Hawkins
1
Hypoth. 1. Bodies move uniformly in straight lines unless so far
as they are retarded by the resistence of yethe Medium or disturbed by
some other force.
Hyp. 2. The alteration of motion is ever proportional to yethe
force by wchwhich it is altered.
Hyp. 3. Two Motions imprest in two different lines, if those lines be taken
in proportion to the motions & completed into a parallelogram, compose a motion whereby the diagonal of yethe Parallelogram shall be
described in the same time in wchwhichyethe sides thereof would have
been described by those compounding motions apart. The
motions AB & AC compound the motion AD.
Prop. 1.
If a body move in vacuo & be continually attracted toward
an immoveable center, it shall constantly move in one & the
same plane, & in that plane describe equal areas in equall
times.
Let A be yethe center towards wchwhichyethe body
is attracted, & suppose yethe attraction acts not
continually but by discontinued impressions
made at equal intervalls of time wchwhich
intervalls we will consider as physical
moments. Let BC be yethe right line in wchwhich
it begins to move from B & wchwhich it describes wthwith uniform
motion in the first physical moment before yethe attraction
make its first impression upon it. At C let it be attracted
towards yethe center A wthwithby one impuls or impression of force, &
let CD be yethe line in wchwhich it shall move after that impuls.
Produce BC to I so that CI be equall to BC & draw ID
parallel to CA & the point D in wchwhich it cuts CD shall be yethe
place of yethe body at the end of yethe second moment. And because
the bases BC CI of the triangles ABC, ACI are equal those
two triangles shall be equal. Also because the triangles ACI, ACD
stand upon the same base AC & between two parallels they shall
be equall. And therefore the triangles ACD described in the second
moment shall be equal to yethe triangle ABD described in the first
moment. And by the same reason if the body at yethe end of the
2d, 3d, 4th, 5t & following moments be attracted by single impulses in D, D, E, F, G &c describing the line DE in yethe 3d moment, EF in
the 4th, FG in yethe 5t &c: the triangle AED shall be equall
to the triangle ADC & all the following triangles AFE, AGF &
to the preceding ones & to one another. And by consequence
the areas compounded of these equall triangles (as ABE, AEG,
ABG &c) are to one another as the lines times in wchwhich they
are described. Suppose now that the moments of time be diminished in length & encreased in number in infinitum, so ytthat
the impulses or impressions of yethe attraction may become continuall & that yethe line BCDEFG by yethe infinite number &
infinite littleness of its sides BC, CD, DE &c may become a
curve one: & the body by the continual attraction shall describe areas of this Curve ABE, AEG, ABG & proportionall to
the times in wchwhich they are described. W. W. to be Dem.
Prop. 2.
If a body be attracted towards either focus of an Ellipsis
& the quantity of the attraction be such as suffices to make yethe
body revolve in the circumference of the Ellipsis: the attraction at yethe two ends of the Ellipsis shall be reciprocally as the
squares of the body in those ends from that focus.
Let AECD be the Ellipsis, A, C its two
ends or vertices, F that focus towards wchwhich
the body is attracted, & AFE, CFD
areas wchwhich the body with a ray drawn
from that focus to its center, describes
at both ends in equal times: & those areas
by the foregoing Proposition must be equal because proportionall to the times: that is the rectangles & must be equal supposing the arches AE & CD
to be so very short that they may be taken for right lines
& therefore AE is to CD as FC to FA. Suppose now
that AM & CN are tangents to the Ellipsis at its two ends
A & C & that EM & DN are perpendiculars let fall from the
points E & D upon those tangents: & because the Ellipsis is
alike crooked at both ends those perpendiculars EM & DN will
be to one another as the squares of the arches AE & CD, &
therefore EM is to DN as FCq to FAq. Now in the times
that the body by means of the attraction moves in the arches AE &2 & CD from A to E & from C to D it would without attraction
move in the tangents from A to M & from C to N. Tis by yethe
force of the attractions that the bodies are drawn out of the tangents from M to E & from N to D & therefore the attractions
are as those distances ME & ND, that is the attraction at the end of the Ellipsis A
is to the attraction at yethe other end of yethe Ellipsis C as ME to
ND & by consequence as FCq to FAq. W. w. to be dem.
Lemma. 1.If a right line touch an Ellipsis in any point thereof &
parallel to that tangent be drawn another right line from the
center of the Ellipsis wchwhich shall intersect a third right line
drawn from yethe touch point through either focus of the Ellipsis: the segment of the last named right line lying between yethe
point of intersection & yethe point of contact shall be equal to
half yethe long axis of yethe Ellipsis.Let APBQ be the Ellipsis; AB its
long axis; C its center; F, f its Foci;
P the point of contact; PR the tangent;
CD the line parallel to the tangent,
& PD the segment of the line FP.
I say that this segment shall be equal
to AC.For joyn PF Pf & draw fE parallel
to CD & because Ff & Fareis bisected in C, & FE shall
be bisected in D & therefore 2PD shall be equal to half the
summ of PF & PE that is to half the summ of PF & Pf, that
is to AB & therefore PD shall be equal to AC. W. W. to be
Dem.Lemma. 2.Every line drawn through either Focus of any Ellipsis &
terminated at both ends by the Ellipsis is to that diameter of the
Ellipsis wchwhich is parallel to this line as the same Diameter is to
the long Axis of the Ellipsis.Let APBQ be yethe Ellipsis, AB its long Axis, F, f its foci, C
its center, PQ yethe line drawn through its focus F, & VCS its
diameter parallel to PQ & PQ will be to VS as VS to AB.For draw FPfp parallel to QFP & cutting the Ellipsis in p.
Joyn Pp cutting VS in T & draw PR wchwhich shall touch the Ellipsis Ellipsis in P & cut the diameter VS produced in R & CT
will be to CS as CS to CR, as has been shewed by all those
who treat of yethe Conic sections. But CT is yethe semisumm of FP
& fp that is of FP & FQ & therefore 2CT is equal to
PQ. Also 2CS is equal to VS & (by yethe foregoing Lemma) 2CR
is equal to AB. Wherefore PQ is to VS as VS to AB. W. W.
to be Dem.Corol. .Lem. 3.If from either focus F of any Ellipsis unto any point in
the perimeter of the Ellipsis be drawn a right line & another
right line doth touch yethe Ellipsis in that point & the angle
of contact be subtended by any third right line drawn parallel
to the first line: the rectangle wchwhich that subtense conteins wthwith
the same subtense produced to the other side of the Ellipsis is to
the rectangle wchwhich the long Axis of the Ellipsis conteins wthwithyethe
first line produced to the other side of the Ellipsis as the
square of the distance between the subtense & the first line is
to the square of the short Axis of the Ellipsis.Let AKBL be the Ellipsis, AB
its long Axis, KL its short Axis, C its
center, F, f its foci, P yethe point of
the perimeter, PF yethe first line PQ
that line produced to the other side
of the Ellipsis PX the tangent, XY yethe
subtense produced to yethe other side of
the Ellipsis & YZ the distance between
this subtense & the first line. I say that
the rectangle YXI is to the rectangle as YZq to KLqFor let VS be the diameter of the Ellipsis parallel
to the first line PF & GH another diametrer parallel to yethe
tangent PX, & the rectangle YXI will be to the square of
the tangent PXq as the rectangle SCV to yethe rectangle GCH
that is as SVq to GHq. This a property of the Ellipsis demonstrated by all that write of the conic sections. And they
have also demonstrated that all the Parallelogramms circumscribed about an Ellipsis are equall. Whence the rectangle is equal to yethe rectangle & consequently GH is to
KL as AB that is (by Lem. 1) 2PD to 2PE & in the same
proportion is PX to YZ. Whence GH PX is to GH as YZ to KL
& PXq to GHq as YZq to KLq. But PXq was to GHq asYXI3 YXI was to PXq as SVq that is (by Lem Cor. Lem. 2) to
GHq, whence invertedly YXI is to as PXq to GHq & by
consequence as YZq to KLq. W. w. to be Dem.Prop. III.If a body be attracted towards either focus of any Ellipsis & by that attraction be made to revolve in the Perimeter
of yethe Ellipsis: the attraction shall be reciprocally as the square
of the distance of the body from that focus of the Ellipsis.Let P be the place of the body at any in the Ellipsis
at any moment of time & PX the tangent in wchwhich the body
would move uniformly were it not attracted & X yethe place
in that tangent at wchwhich it would arrive in any given part
of time & Y the place in the perimeter of the Ellipsis
at wchwhich the body doth arrive in the same time by means of
the attraction. Let us suppose the time to be divided into
equal parts & that those parts are very little ones so ytthat
they may be considered as physical moments & ytthatyethe attraction acts not continually but by intervalls only once in the beginning of every physical moment & let yethe first action be
upon yethe body in P, the next upon it in Y & so on perpetually, so ytthatyethe body may move from P to Y in the chord
of yethe arch PY & from Y to its next place in yethe Ellipsis in the chord of yethe next arch & so on for ever. And
because the attraction in P is made towards F & diverts
the body from yethe tangent PX into yethe chord PY so that
in the end of the first physical moment it be not found
in the place X where it would have been without yethe attraction but in Y being by yethe force of yethe attraction in P
translated from X to Y: the line XY generated by the
force of yethe attraction in P must be proportional to that
force & parallel to its direction that is parallel to PF
Produce XY & PF till they cut the Ellipsis
in I & Q. Ioyn FY & upon FP let fall
the perperpendicularperpendicular YZ & let AB be the
long Axis & KL yethe short Axis of yethe
Ellipses. And by the third Lemma YXI
will be to as YZq to KLq
& by consequence YX will be equall to .And in like manner if py be the chord of another Arch
py wchwhich the revolving body describes in a physical moment of time
& px be the tangent of the Ellipsis at p & xy the subtense of the the angle of contact drawn parallel to pF, & if pF & xy produced cut yethe Ellipsis in q & i & from y upon pF be let fall the
perpendicular yz: the subtense yx shall be equal to .
And therefore YX shall be to yx as to ,
that is as to And because the lines PY py are by the revolving body
described in equal times, the areas of the triangles PYF pyF
must be equal by the first Proposition; & therefore the rectangles & are equal, & by consequence YZ is
to yz as pF to PF. Whence is to as to And therefore YX is to yx as
to .And as we told you that XY was the line generated in
a physical moment of time by yethe force of the attraction in P,
so for the same reason is xy the line generated in the
same quantity of time by the force of the attraction in p.
And therefore the attraction in P is to the attraction in p
as the line XY to the line xy, that is as to Suppose now that the equal lines in wchwhich the revolving
body describes the lines PY & py become infinitely little, so
that the attraction may become continual & the body by this
attraction revolve in the perimeter of the Ellipsis: & the lines
PQ, XI as also pq, xi becoming coincident & by consequence
equal, the quantities & will become
& . And therefore the attraction in P will be to
the attraction in p as to , that is reciprocally as
the squares of the distances of the revolving bodies from the
focus of the Ellipsis. W. W. to be Dem.