Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <p type="main">
                  <s>
                    <pb xlink:href="039/01/154.jpg" pagenum="126"/>
                    <arrow.to.target n="note102"/>
                  curvam
                    <emph type="italics"/>
                  Vpk.
                    <emph.end type="italics"/>
                  Eſt autem hæc Curva
                    <emph type="italics"/>
                  Vpk
                    <emph.end type="italics"/>
                  eadem cum Curva illa
                    <lb/>
                    <emph type="italics"/>
                  VPQ
                    <emph.end type="italics"/>
                  in Corol. </s>
                  <s>3. Prop. </s>
                  <s>XLI inventa, in qua ibi diximus corpora
                    <lb/>
                  hujuſmodi viribus attracta oblique aſcendere. </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note102"/>
                  DE MOTU
                    <lb/>
                  CORPORUM</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  PROPOSITIO XLV. PROBLEMA XXXI.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                    <emph type="italics"/>
                  Orbium qui ſunt Circulis maxime finitimi requiruntur motus Ap­
                    <lb/>
                  ſidum.
                    <emph.end type="italics"/>
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>Problema ſolvitur Arithmetice faciendo ut Orbis, quem corpus
                    <lb/>
                  in Ellipſi mobili (ut in Propoſitionis ſuperioris Corol. </s>
                  <s>2, vel 3)
                    <lb/>
                  revolvens deſcribit in plano immobili, accedat ad formam Orbis
                    <lb/>
                  cujus Apſides requiruatur, & quærendo Apſides Orbis quem cor­
                    <lb/>
                  pus illud in plano immobili deſcribit. </s>
                  <s>Orbes autem eandem ac­
                    <lb/>
                  quirent formam, ſi vires centripetæ quibus deſcribuntur, inter ſe
                    <lb/>
                  collatæ, in æqualibus altitudinibus reddantur proportionales. </s>
                  <s>Sit
                    <lb/>
                  punctum
                    <emph type="italics"/>
                  V
                    <emph.end type="italics"/>
                  Apſis ſumma, & ſcribantur T pro altitudine maxima
                    <lb/>
                    <emph type="italics"/>
                  CV,
                    <emph.end type="italics"/>
                  A pro altitudine quavis alia
                    <emph type="italics"/>
                  CP
                    <emph.end type="italics"/>
                  vel
                    <emph type="italics"/>
                  Cp,
                    <emph.end type="italics"/>
                  & X pro alti­
                    <lb/>
                  titudinum differentia
                    <emph type="italics"/>
                  CV-CP
                    <emph.end type="italics"/>
                  ; & vis qua corpus in Ellipſi
                    <lb/>
                  circa umbilicum ſuum
                    <emph type="italics"/>
                  C
                    <emph.end type="italics"/>
                  (ut in Corollario 2.) revolvente move­
                    <lb/>
                  tur, quæQ.E.I. Corollario 2. erat ut (FF/AA)+(RGG-RFF/A
                    <emph type="italics"/>
                  cub.
                    <emph.end type="italics"/>
                  ), id eſt
                    <lb/>
                  ut (FFA+RGG-RFF/A
                    <emph type="italics"/>
                  cub.
                    <emph.end type="italics"/>
                  ), ſubſtituendo T-X pro A, erit ut
                    <lb/>
                  (RGG-RFF+TFF-FFX/A
                    <emph type="italics"/>
                  cub.
                    <emph.end type="italics"/>
                  ). Reducenda ſimiliter eſt vis alia
                    <lb/>
                  quævis centripeta ad fractionem cujus denominator ſit A
                    <emph type="italics"/>
                  cub.,
                    <emph.end type="italics"/>
                  &
                    <lb/>
                  numeratores, facta homologorum terminorum collatione, ſtatuendi
                    <lb/>
                  ſunt analogi. </s>
                  <s>Res Exemplis patebit. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Exempl.
                    <emph.end type="italics"/>
                  1. Ponamus vim centripetam uniformem eſſe, adeoque
                    <lb/>
                  ut (A
                    <emph type="italics"/>
                  cub.
                    <emph.end type="italics"/>
                  /A
                    <emph type="italics"/>
                  cub.
                    <emph.end type="italics"/>
                  ), ſive (ſcribendo T-X pro A in Numeratore) ut
                    <lb/>
                  (T
                    <emph type="italics"/>
                  cub.
                    <emph.end type="italics"/>
                  -3TTX+3TXX-X
                    <emph type="italics"/>
                  cub.
                    <emph.end type="italics"/>
                  /A
                    <emph type="italics"/>
                  cub.
                    <emph.end type="italics"/>
                  ); & collatis Numeratorum ter­
                    <lb/>
                  minis correſpondentibus, nimirum datis cum datis & non datis
                    <lb/>
                  cum non datis, fiet RGG-RFF+TFF ad T
                    <emph type="italics"/>
                  cub.
                    <emph.end type="italics"/>
                  ut-FFX ad
                    <lb/>
                  -3TTX+3TXX-X
                    <emph type="italics"/>
                  cub.
                    <emph.end type="italics"/>
                  ſive ut-FF ad-3TT+3TX
                    <lb/>
                  -XX. </s>
                  <s>Jam cum Orbis ponatur Circulo quam maxime finitimus,
                    <lb/>
                  coeat Orbis cum Circulo; & ob factas R, T æquales, atque X in infi-</s>
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