Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                  <s>
                    <pb xlink:href="039/01/284.jpg" pagenum="256"/>
                    <arrow.to.target n="note232"/>
                  tam in eodem loco ut 1/2
                    <emph type="italics"/>
                  OS
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  OP.
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                  Nam vires illæ ſunt ad invi­
                    <lb/>
                  vicem ut 1/4
                    <emph type="italics"/>
                  Rr
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                  &
                    <emph type="italics"/>
                  TQ
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                  ſive ut (1/4
                    <emph type="italics"/>
                  VQXPQ/SQ
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                  ) & (1/2
                    <emph type="italics"/>
                  PQq/SP
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                  ), hoc eſt, ut 1/2
                    <emph type="italics"/>
                  VQ
                    <emph.end type="italics"/>
                    <lb/>
                  &
                    <emph type="italics"/>
                  PQ,
                    <emph.end type="italics"/>
                  ſeu 1/2
                    <emph type="italics"/>
                  OS
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                  &
                    <emph type="italics"/>
                  OP.
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                  Data igitur Spirali datur proportio re­
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                  ſiſtentiæ ad vim centripetam, & viceverſa ex data illa proportione
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                  datur Spiralis. </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note232"/>
                  DE MOTU
                    <lb/>
                  CORPORUM</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  4. Corpus itaque gyrari nequit in hac Spirali, niſi ubi vis
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                  reſiſtentiæ minor eſt quam dimidium vis centripetæ. </s>
                  <s>Fiat reſiſten­
                    <lb/>
                  tia æqualis dimidio vis centripetæ & Spiralis conveniet cum linea
                    <lb/>
                  recta
                    <emph type="italics"/>
                  PS,
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                  inque hac recta corpus deſcendet ad centrum, ea cum
                    <lb/>
                  velocitate quæ ſit ad velocitatem qua probavimus in ſuperioribus
                    <lb/>
                  in caſu Parabolæ (Theor. </s>
                  <s>X, Lib. </s>
                  <s>I,) deſcenſum in Medio non reſi­
                    <lb/>
                  ſtente fieri, in ſubduplicata ratione unitatis ad numerum binarium. </s>
                  <s>
                    <lb/>
                  Et tempora deſcenſus hic erunt reciproce ut velocitates, atque
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                  adeo dantur. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  5. Et quoniam in æqualibus a centro diſtantiis velocitas
                    <lb/>
                  eadem eſt in Spirali
                    <emph type="italics"/>
                  PQR
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                  atQ.E.I. recta
                    <emph type="italics"/>
                  SP,
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                  & longitudo Spi­
                    <lb/>
                  ralis ad longitudinem rectæ
                    <emph type="italics"/>
                  PS
                    <emph.end type="italics"/>
                  eſt in data ratione, nempe in
                    <lb/>
                  ratione
                    <emph type="italics"/>
                  OP
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  OS
                    <emph.end type="italics"/>
                  ; tempus deſcenſus in Spirali erit ad tem­
                    <lb/>
                  pus deſcenſus in recta
                    <emph type="italics"/>
                  SP
                    <emph.end type="italics"/>
                  in eadem illa data ratione, proinde­
                    <lb/>
                  Q.E.D.tur. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  6. Si centro
                    <emph type="italics"/>
                  S
                    <emph.end type="italics"/>
                  intervallis duobus quibuſcunQ.E.D.tis deſcri­
                    <lb/>
                  bantur duo Circuli; & manentibus hiſce Circulis, mutetur utcun­
                    <lb/>
                  que angulus quem Spiralis continet cum radio
                    <emph type="italics"/>
                  PS:
                    <emph.end type="italics"/>
                  numerus revo­
                    <lb/>
                  lutionum quas corpus intra Circulorum circumferentias, pergendo
                    <lb/>
                  in Spirali a circumferentia ad circumferentiam, complere poteſt, eſt
                    <lb/>
                  ut (
                    <emph type="italics"/>
                  PS/OS
                    <emph.end type="italics"/>
                  ), ſive ut Tangens anguli illius quem Spiralis continet cum
                    <lb/>
                  radio
                    <emph type="italics"/>
                  PS
                    <emph.end type="italics"/>
                  ; tempus vero revolutionum earundem ut (
                    <emph type="italics"/>
                  OP/OS
                    <emph.end type="italics"/>
                  ), id eſt, ut
                    <lb/>
                  Secans anguli ejuſdem, vel etiam reciproce ut Medii denſitas. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  7. Si corpus, in Medio cujus denſitas eſt reciproce ut di­
                    <lb/>
                  ſtantia loeorum a centro, revolutionem in Curva quacunque
                    <emph type="italics"/>
                  AEB
                    <emph.end type="italics"/>
                    <lb/>
                  circa centrum illud fecerit, & Radium primum
                    <emph type="italics"/>
                  AS
                    <emph.end type="italics"/>
                  in eodem an­
                    <lb/>
                  gulo ſecuerit in
                    <emph type="italics"/>
                  B
                    <emph.end type="italics"/>
                  quo prius in
                    <emph type="italics"/>
                  A,
                    <emph.end type="italics"/>
                  idque cum velocitate quæ fue­
                    <lb/>
                  rit ad velocitatem ſuam primam in
                    <emph type="italics"/>
                  A
                    <emph.end type="italics"/>
                  reciproce in ſubduplica­
                    <lb/>
                  ta ratione diſtantiarum a centro (id eſt, ut
                    <emph type="italics"/>
                  AS
                    <emph.end type="italics"/>
                  ad mediam pro­
                    <lb/>
                  portionalem inter
                    <emph type="italics"/>
                  AS
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  BS
                    <emph.end type="italics"/>
                  ) corpus illud perget innume­
                    <lb/>
                  ras conſimiles revolutiones
                    <emph type="italics"/>
                  BFC, CGD
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>facere, & interſe-</s>
                </p>
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