Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                  DE MOTU
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                  CORPORUM</s>
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                  PROPOSITIO XV. THEOREMA XII.
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                  Si Medii denſitas in locis ſingulis ſit reciproce ut diſtantia loeorum
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                  a centro immobili, ſitque vis centripeta in duplicata ratione den­
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                  ſitatis: dico quod corpus gyrari potest in Spirali, quæ radios
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                  omnes a centro illo ductos interſecat in angulo dato.
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                  <s>Ponantur quæ in ſuperiore Lemmate, & producatur
                    <emph type="italics"/>
                  SQ
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                  ad
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                  V,
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                    <lb/>
                  ut ſit
                    <emph type="italics"/>
                  SV
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                  æqualis
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                  SP.
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                  Tempore quovis, in Medio reſiſtente, de­
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                  ſcribat corpus arcum quam minimum
                    <emph type="italics"/>
                  PQ,
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                  & tempore duplo ar­
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                  cum quam minimum
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                  PR
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                  ; & decrementa horum arcuum ex reſi­
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                  ſtentia oriunda, ſive defe­
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                  ctus ab arcubus qui in Me­
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                  dio non reſiſtente iiſdem
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                  temporibus deſcriberen­
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                  tur, erunt ad invicem ut
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                  quadrata temporum in
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                  quibus generantur: Eſt
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                  itaQ.E.D.crementum arcus
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                    <emph type="italics"/>
                  PQ
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                  pars quarta decre­
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                  menti arcus
                    <emph type="italics"/>
                  PR.
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                  Unde
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                  etiam, ſi areæ
                    <emph type="italics"/>
                  PSQ
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                  æ­
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                  qualis capiatur area
                    <emph type="italics"/>
                  QSr,
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                  erit decrementum arcus
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                    <emph type="italics"/>
                  PQ
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                  æquale dimidio lineolæ
                    <emph type="italics"/>
                  Rr
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                  ; adeoque vis reſiſtentiæ & vis cen­
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                  tripeta ſunt ad invicem ut lineolæ 1/2
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                  Rr
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                  &
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                  TQ
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                  quas ſimul generant. </s>
                  <s>
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                  Quoniam vis centripeta, qua corpus urgetur in
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                  P,
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                  eſt reciproce ut
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                    <emph type="italics"/>
                  SPq,
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                  & (per Lem. </s>
                  <s>X. Lib. </s>
                  <s>1,) lineola
                    <emph type="italics"/>
                  TQ,
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                  quæ vi illa generatur, eſt
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                  in ratione compoſita ex ratione hujus vis & ratione duplicata tem­
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                  poris quo arcus
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                  PQ
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                  deſcribitur, (Nam reſiſtentiam in hoc caſu,
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                  ut infinite minorem quam vis centripeta, negligo) erit
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                  TQXSPq
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                  id eſt (per Lemma noviſſimum) 1/2
                    <emph type="italics"/>
                  PQqXSP,
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                  in ratione duplicata
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                  temporis, adeoque tempus eſt ut
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                  PQX√SP
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                  ; & corporis veloci­
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                  tas, qua arcus
                    <emph type="italics"/>
                  PQ
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                  illo tempore deſcribitur, ut (
                    <emph type="italics"/>
                  PQ/PQX√SP
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                  ) ſeu
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                  (1/√
                    <emph type="italics"/>
                  SP
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                  ), hoc eſt, in ſubduplicata ratione ipſius
                    <emph type="italics"/>
                  SP
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                  reciproce. </s>
                  <s>Et ſi­
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                  mili argumento, velocitas qua arcus
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                  QR
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                  deſcribitur, eſt in ſub-</s>
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