Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <p type="main">
                  <s>
                    <pb xlink:href="039/01/082.jpg" pagenum="54"/>
                    <arrow.to.target n="note30"/>
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                <p type="margin">
                  <s>
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                  DE MOTU
                    <lb/>
                  CORPORUM</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  Hinc Ellipſeos area tota, eique proportionale rectangu­
                    <lb/>
                  lum ſub axibus, eſt in ratione compoſita ex ſubduplicata ratione
                    <lb/>
                  lateris recti & ratione temporis periodici. </s>
                  <s>Namque area tota eſt
                    <lb/>
                  ut area
                    <emph type="italics"/>
                  QTXSP,
                    <emph.end type="italics"/>
                  quæ dato tempore deſcribitur, ducta in &c. </s>
                  <s>ducta in tempus periodicum. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  PROPOSITIO XV. THEOREMA VII.
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                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                    <emph type="italics"/>
                  Iiſdem poſitis, dico quod Tempora periodica in Ellipſibus ſunt in ratione
                    <lb/>
                  ſeſquiplicata majorum axium.
                    <emph.end type="italics"/>
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>Namque axis minor eſt medius proportionalis inter axem majo­
                    <lb/>
                  rem & latus rectum, atque adeo rectangulum ſub axibus eſt in ra­
                    <lb/>
                  tione compoſita ex ſubduplicata ratione lateris recti & ſeſquiplicata
                    <lb/>
                  ratione axis majoris. </s>
                  <s>Sed hoc rectangulum, per Corollarium Prop. </s>
                  <s>
                    <lb/>
                  XIV. eſt in ratione compoſita ex ſubduplicata ratione lateris recti
                    <lb/>
                  & ratione periodici temporis. </s>
                  <s>Dematur utrobique ſubduplicata
                    <lb/>
                  ratio lateris recti, & manebit ſeſquiplicata ratio majoris axis æqua­
                    <lb/>
                  lis rationi periodici temporis.
                    <emph type="italics"/>
                  Q.E.D.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  Sunt igitur tempora periodica in Ellipſibus eadem ac in
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                  Circulis, quorum diametri æquantur majoribus axibus Ellipſeon. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="center"/>
                  PROPOSITIO XVI. THEOREMA VIII.
                    <emph.end type="center"/>
                  </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Iiſdem poſitis, & actis ad corpora lineis rectis, quæ ibidem tangant Or­
                    <lb/>
                  bitas, demiſſiſque ab umbilico communi ad has tangentes perpendi­
                    <lb/>
                  cularibus: dico quod Velocitates corporum ſunt in ratione compoſi­
                    <lb/>
                  ta ex ratione perpendiculorum inverſe & ſubduplicata ratione la­
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                  terum rectorum principalium directe.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="main">
                  <s>Ab umbilico
                    <emph type="italics"/>
                  S
                    <emph.end type="italics"/>
                  ad tangentem
                    <emph type="italics"/>
                  PR
                    <emph.end type="italics"/>
                  demitte perpendiculum
                    <emph type="italics"/>
                  SY
                    <emph.end type="italics"/>
                    <lb/>
                  & velocitas corporis
                    <emph type="italics"/>
                  P
                    <emph.end type="italics"/>
                  erit reciproce in ſubduplicata ratione quan­
                    <lb/>
                  titatis (
                    <emph type="italics"/>
                  SYq/L
                    <emph.end type="italics"/>
                  ). Nam velocitas illa eſt ut arcus quam minimus
                    <emph type="italics"/>
                  PQ
                    <emph.end type="italics"/>
                    <lb/>
                  in data temporis particula deſcriptus, hoc eſt (per Lem. </s>
                  <s>VII.) ut
                    <lb/>
                  tangens
                    <emph type="italics"/>
                  PR,
                    <emph.end type="italics"/>
                  id eſt (ob proportionales
                    <emph type="italics"/>
                  PR
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  QT
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  SP
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  SY
                    <emph.end type="italics"/>
                  ) ut
                    <lb/>
                  (
                    <emph type="italics"/>
                  SPXQT/SY
                    <emph.end type="italics"/>
                  ), ſive ut
                    <emph type="italics"/>
                  SY
                    <emph.end type="italics"/>
                  reciproce &
                    <emph type="italics"/>
                  SPXQT
                    <emph.end type="italics"/>
                  directe; eſtque </s>
                </p>
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