Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                  <s>
                    <pb xlink:href="039/01/257.jpg" pagenum="229"/>
                  tatem illam datam in ſubduplicata ratione, quam habet vis Gravi­
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                  tatis ad Medii reſiſtentiam illam cognitam. </s>
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                <p type="margin">
                  <s>
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                  LIBER
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                  SECUMDUS.</s>
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                <p type="main">
                  <s>
                    <emph type="center"/>
                  PROPOSITIO IX. THEOREMA VII.
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                  </s>
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                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Poſitis jam demonſtratis, dico quod ſi Tangentes angulorum ſecto
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                  ris Circularis & ſectoris Hyperbolici ſumantur velocitatibus
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                  proportionales, exiſtente radio juſtæ magnitudinis: erit tempus
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                  omne aſcenſus futuri ut ſector Circuli, & tempus omne deſcen­
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                  ſus præteriti ut ſector Hyperbolæ.
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                  </s>
                </p>
                <p type="main">
                  <s>Rectæ
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                  AC,
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                  qua vis gravitatis exponitur, perpendicularis & æ­
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                  qualis ducatur
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                  AD.
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                  Centro
                    <emph type="italics"/>
                  D
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                  ſemidiametro
                    <emph type="italics"/>
                  AD
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                  deſcribatur tum
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                  Circuli quadrans
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                  AtE,
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                  tum Hyperbola rectangula
                    <emph type="italics"/>
                  AVZ
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                  axem
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                  habens
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                  AX,
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                  verticem principalem
                    <emph type="italics"/>
                  A
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                  & Aſymptoton
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                  DC.
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                  Jun­
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                  gantur
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                  Dp, DP,
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                  & erit ſector Circularis
                    <emph type="italics"/>
                  AtD
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                  ut tempus aſcenſus
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                  omnis futuri; & ſector Hyperbolicus
                    <emph type="italics"/>
                  ATD
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                  ut tempus deſcenſus
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                  omnis præteriti. </s>
                  <s>Si modo ſectorum Tangentes
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                  Ap, AP
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                  ſint ut
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                  velocitates. </s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Cas.
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                  1. Agatur enim
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                  Dvq
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                  abſcindens ſectoris
                    <emph type="italics"/>
                  ADt
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                  & trian­
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                  guli
                    <emph type="italics"/>
                  ADp
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                  momenta, ſeu particulas quam minimas ſimul deſcrip­
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                  tas
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                  tDv
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                  &
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                    <expan abbr="pDq.">pDque</expan>
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                  Cum particulæ illæ, ob angulum commu­
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                  nem
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                  D,
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                  ſunt in duplicata ratione laterum, erit particula
                    <emph type="italics"/>
                  tDv
                    <emph.end type="italics"/>
                    <lb/>
                  ut (
                    <emph type="italics"/>
                  qDp/pDquad
                    <emph.end type="italics"/>
                  ). Sed
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                  pDquad.
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                  eſt
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                  ADquad+Apquad.
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                  id eſt,
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                    <emph type="italics"/>
                  ADquad+ADXAk
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                  ſeu
                    <emph type="italics"/>
                  ADXCk
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                  ; &
                    <emph type="italics"/>
                  qDp
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                  eſt 1/2
                    <emph type="italics"/>
                    <expan abbr="ADXpq.">ADXpque</expan>
                    <emph.end type="italics"/>
                    <lb/>
                  Ergo ſectoris particula
                    <emph type="italics"/>
                  tDv
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                  eſt ut (
                    <emph type="italics"/>
                  pq/Ck
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                  ), id eſt, ut velocitatis de­
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                  crementum quam minimum
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                  pq
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                  directe & vis illa
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                  Ck
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                  quæ velo­
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                  citatem diminuit inverſe, atque adeo ut particula temporis decre­
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                  mento reſpondens. </s>
                  <s>Et componendo fit ſumma particularum om­
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                  nium
                    <emph type="italics"/>
                  tDv
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                  in ſectore
                    <emph type="italics"/>
                  ADt,
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                  ut ſumma particularum temporis
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                  ſingulis velocitatis decreſcentis
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                  Ap
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                  particulis amiſſis
                    <emph type="italics"/>
                  pq
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                  reſpon­
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                  dentium, uſQ.E.D.m velocitas illa in nihilum diminuta eva­
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                  nuerit; hoc eſt, ſector totus
                    <emph type="italics"/>
                  ADt
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                  eſt ut aſcenſus totius futuri
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                  tempus.
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                  Q.E.D.
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                  </s>
                </p>
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