Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <p type="main">
                  <s>
                    <pb xlink:href="039/01/158.jpg" pagenum="130"/>
                    <arrow.to.target n="note106"/>
                  dex eſt (
                    <emph type="italics"/>
                  nn/mm
                    <emph.end type="italics"/>
                  )-3. Id quod per Exempla ſecunda manifeſtum eſt. </s>
                  <s>
                    <lb/>
                  Unde liquet vim illam in majore quam triplicata altitudinis ratione,
                    <lb/>
                  in receſſu a centro, decreſcere non poſſe: Corpus tali vi revolvens
                    <lb/>
                  deque Apſide diſcedens, ſi cæperit deſcendere nunquam perveniet
                    <lb/>
                  ad Apſidem imam ſeu altitudinem minimam, ſed deſcendet uſque ad
                    <lb/>
                  centrum, deſcribens Curvam illam lineam de qua egimus in Cor. </s>
                  <s>3.
                    <lb/>
                  Prop. </s>
                  <s>XLI. </s>
                  <s>Sin cæperit illud, de Apſide diſcedens, vel minimum
                    <lb/>
                  aſcendere; aſcendet in infinitum, neque unquam perveniet ad Ap­
                    <lb/>
                  ſidem ſummam. </s>
                  <s>Deſcribet enim Curvam illam lineam de qua ac­
                    <lb/>
                  tum eſt in eodem Corol. </s>
                  <s>& in Corol. </s>
                  <s>6, Prop. </s>
                  <s>XLIV. </s>
                  <s>Sic & ubi
                    <lb/>
                  vis, in receſſu a centro, decreſcit in majore quam triplicata ratione
                    <lb/>
                  altitudinis, corpus de Apſide diſcedens, perinde ut cæperit deſcen­
                    <lb/>
                  dere vel aſcendere, vel deſcendet ad centrum uſque vel aſcendet
                    <lb/>
                  in infinitum. </s>
                  <s>At ſi vis, in receſſu a centro, vel decreſcat in minore
                    <lb/>
                  quam triplicata ratione altitudinis, vel creſcat in altitudinis ratione
                    <lb/>
                  quacunque; corpus nunquam deſcendet ad centrum uſque, ſed ad
                    <lb/>
                  Apſidem imam aliquando perveniet: & contra, ſi corpus de Apſi­
                    <lb/>
                  de ad Apſidem alternis vicibus deſcendens & aſcendens nunquam
                    <lb/>
                  appellat ad centrum; vis in receſſu a centro aut augebitur, aut in
                    <lb/>
                  minore quam triplicata altitudinis ratione decreſcet: & quo ci­
                    <lb/>
                  tius corpus de Apſide ad Apſidem redierit, eo longius ratio virium
                    <lb/>
                  recedet a ratione illa triplicata. </s>
                  <s>Ut ſi corpus revolutionibus 8 vel
                    <lb/>
                  4 vel 2 vel 1 1/2 de Apſide ſumma ad Apſidem ſummam alterno de­
                    <lb/>
                  ſcenſu & aſcenſu redierit; hoc eſt, ſi fuerit
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  ut 8 vel 4 vel
                    <lb/>
                  2 vel 1 1/2 ad 1, adeoque (
                    <emph type="italics"/>
                  nn/mm
                    <emph.end type="italics"/>
                  )-3 valeat (1/64)-3 vel (1/16) -3 vel 1/4-3
                    <lb/>
                  vel 4/9-3: erit vis ut A
                    <emph type="sup"/>
                  (1/64)-3
                    <emph.end type="sup"/>
                  vel A
                    <emph type="sup"/>
                  (1/16)-3
                    <emph.end type="sup"/>
                  vel A
                    <emph type="sup"/>
                  1/4-3
                    <emph.end type="sup"/>
                  vel A
                    <emph type="sup"/>
                  4/9-3
                    <emph.end type="sup"/>
                  ,
                    <lb/>
                  id eſt, reciproce ut A
                    <emph type="sup"/>
                  3-(1/64)
                    <emph.end type="sup"/>
                  vel A
                    <emph type="sup"/>
                  3-(1/16)
                    <emph.end type="sup"/>
                  vel A
                    <emph type="sup"/>
                  3-1/4
                    <emph.end type="sup"/>
                  vel A
                    <emph type="sup"/>
                  3-4/9
                    <emph.end type="sup"/>
                  . </s>
                  <s>
                    <lb/>
                  Si corpus ſingulis revolutionibus redierit ad Apſidem eandem immo­
                    <lb/>
                  tam; erit
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  ut 1 ad 1, adeoque A (
                    <emph type="italics"/>
                  nn/mm
                    <emph.end type="italics"/>
                  )-3 æqualis A
                    <emph type="sup"/>
                  -2
                    <emph.end type="sup"/>
                  ſeu (1/AA
                    <gap/>
                  )
                    <lb/>
                  & propterea decrementum virium in ratione duplicata altitudinis,
                    <lb/>
                  ut in præcedentibus demonſtratum eſt. </s>
                  <s>Si corpus partibus revo­
                    <lb/>
                  lutionis unius vel tribus quartis, vel duabus tertiis, vel una ter­
                    <lb/>
                  tia, vel una quarta, ad Apſidem eandem redierit; erit
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  ut
                    <lb/>
                  1/4 vel 2/3 vel 1/3 vel 1/4 ad 1, adeoque A(
                    <emph type="italics"/>
                  nn/mm
                    <emph.end type="italics"/>
                  )-3 æqualis A
                    <emph type="sup"/>
                  (16/9)-3
                    <emph.end type="sup"/>
                  vel
                    <lb/>
                  A
                    <emph type="sup"/>
                  9/4-3
                    <emph.end type="sup"/>
                  vel A
                    <emph type="sup"/>
                  9-3
                    <emph.end type="sup"/>
                  vel A
                    <emph type="sup"/>
                  16-3
                    <emph.end type="sup"/>
                  ; & propterea vis aut reciproce ut </s>
                </p>
              </subchap2>
            </subchap1>
          </chap>
        </body>
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