Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                  <s>
                    <pb xlink:href="039/01/286.jpg" pagenum="258"/>
                    <arrow.to.target n="note234"/>
                  quo Spiralis nova ſecat radium eundem in Medio propoſito: At­
                    <lb/>
                  que etiam ut ſunt eorundem angulorum Tangentes ita eſſe numeros
                    <lb/>
                  revolutionum omnium inter Circulos eoſdem duos quam proxime. </s>
                  <s>
                    <lb/>
                  Si hæc fiant paſſim inter Circulos binos, continuabitur motus per
                    <lb/>
                  Circulos omnes. </s>
                  <s>Atque hoc pacto haud difficulter imaginari poſſi­
                    <lb/>
                  mus quibus modis ac temporibus corpora in Medio quocunque re­
                    <lb/>
                  gulari gyrari debebunt. </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note234"/>
                  DE MOTU
                    <lb/>
                  CORPORUM</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  9. Et quamvis motus excentrici in Spiralibus ad formam
                    <lb/>
                  Ovalium accedentibus peragantur; tamen concipiendo Spiralium
                    <lb/>
                  illarum ſingulas revolutiones iiſdem ab invicem intervallis diſtare,
                    <lb/>
                  iiſdemque gradibus ad centrum accedere cum Spirali ſuperius de­
                    <lb/>
                  ſcripta, intelligemus etiam quomodo motus corporum in hujuſmo­
                    <lb/>
                  di Spiralibus peragantur. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  PROPOSITIO XVI. THEOREMA XIII.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Si Medii denſitas in locis ſingulis ſit reciproce ut diſtantia loco­
                    <lb/>
                  rum a centro immobili, ſitque vis centripeta reciproce ut dig­
                    <lb/>
                  nitas quælibet ejuſdem diſtantiæ: dico quod corpus gyrari potest
                    <lb/>
                  in Spirali quæ radios omnes a centro illo ductos interſecat in
                    <lb/>
                  angulo dato.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>Demonſtratur eadem methodo cum Propoſitione ſuperiore. </s>
                  <s>
                    <lb/>
                  Nam ſi vis centripeta in
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  ſit reciproce ut diſtantiæ
                    <emph type="italics"/>
                  SP
                    <emph.end type="italics"/>
                  dignitas
                    <lb/>
                  quælibet
                    <emph type="italics"/>
                  SP
                    <emph type="sup"/>
                  n
                    <emph.end type="italics"/>
                  +1
                    <emph.end type="sup"/>
                  cujus index eſt
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  +1; colligetur ut ſupra,
                    <lb/>
                  quod tempus quo corpus deſcribit arcum quemvis
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                  erit ut
                    <lb/>
                    <emph type="italics"/>
                  PQXSP
                    <emph.end type="italics"/>
                    <emph type="sup"/>
                  1/2
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                    <emph.end type="sup"/>
                  , & reſiſtentia in
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  ut (
                    <emph type="italics"/>
                  Rr/PQqXSP
                    <emph type="sup"/>
                  n
                    <emph.end type="sup"/>
                    <emph.end type="italics"/>
                  ), ſive ut (—1-1/2
                    <emph type="italics"/>
                  nXVQ/PQXSP
                    <emph type="sup"/>
                  n
                    <emph.end type="sup"/>
                  XSQ
                    <emph.end type="italics"/>
                  ),
                    <lb/>
                  adeoque ut (—1-1/2
                    <emph type="italics"/>
                  nXOS/OPXSP
                    <emph type="sup"/>
                  n+1
                    <emph.end type="sup"/>
                    <emph.end type="italics"/>
                  ), hoc eſt, ob datum (—1-1/2
                    <emph type="italics"/>
                  nXOS/OP
                    <emph.end type="italics"/>
                  ), recipro­
                    <lb/>
                  ce ut
                    <emph type="italics"/>
                  SP
                    <emph type="sup"/>
                  n+1
                    <emph.end type="sup"/>
                  .
                    <emph.end type="italics"/>
                  Et propterea, cum velocitas ſit reciproce ut
                    <emph type="italics"/>
                  SP
                    <emph.end type="italics"/>
                    <emph type="sup"/>
                  1/2
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                    <emph.end type="sup"/>
                  ,
                    <lb/>
                  denſitas in
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  erit reciproce ut
                    <emph type="italics"/>
                  SP.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  1. Reſiſtentia eſt ad vim centripetam, ut —1-1/2
                    <emph type="italics"/>
                  nXOS
                    <emph.end type="italics"/>
                    <lb/>
                  ad
                    <emph type="italics"/>
                  OP.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  2. Si vis centripeta ſit reciproce ut
                    <emph type="italics"/>
                  SPcub,
                    <emph.end type="italics"/>
                  erit 1-1/2
                    <emph type="italics"/>
                  n=o
                    <emph.end type="italics"/>
                  ;
                    <lb/>
                  adeoque reſiſtentia & denſitas Medii nulla erit, ut in Propoſitione
                    <lb/>
                  nona Libri primi. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  3. Si vis centripeta ſit reciproce ut dignitas aliqua radii
                    <lb/>
                    <emph type="italics"/>
                  SP
                    <emph.end type="italics"/>
                  cujus index eſt major numero 3, reſiſtentia affirmativa in nega­
                    <lb/>
                  tivam mutabitur. </s>
                </p>
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