Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                  <s>
                    <pb xlink:href="039/01/156.jpg" pagenum="128"/>
                    <arrow.to.target n="note104"/>
                  ut 1 ad √
                    <emph type="italics"/>
                  n.
                    <emph.end type="italics"/>
                  Quare cum angulus
                    <emph type="italics"/>
                  VCP,
                    <emph.end type="italics"/>
                  in deſcenſu corporis
                    <lb/>
                  ab Apſide ſumma ad Apſidem imam in Ellipſi confectus, ſit
                    <lb/>
                  graduum 180; conficietur angulus
                    <emph type="italics"/>
                  VCp,
                    <emph.end type="italics"/>
                  in deſcenſu corporis
                    <lb/>
                  ab Apſide ſumma ad Apſidem imam, in Orbe propemodum Cir­
                    <lb/>
                  culari quem corpus quodvis vi centripeta dignitati A
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  -3
                    <emph.end type="sup"/>
                  pro­
                    <lb/>
                  portionali deſcribit, æqualis angulo graduum (180/√
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  ); & hoc angulo
                    <lb/>
                  repetito corpus redibit ab Apſide ima ad Apſidem ſummam, &
                    <lb/>
                  ſic deinceps in infinitum. </s>
                  <s>Ut ſi vis centripeta ſit ut diſtantia cor­
                    <lb/>
                  poris a centro, id eſt, ut A ſeu (A
                    <emph type="sup"/>
                  4
                    <emph.end type="sup"/>
                  /A
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  ), erit
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  æqualis 4 & √
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  æqualis 2;
                    <lb/>
                  adeoque angulus inter Apſidem ſummam & Apſidem imam æ­
                    <lb/>
                  qualis (180/2)
                    <emph type="italics"/>
                  gr.
                    <emph.end type="italics"/>
                  ſeu 90
                    <emph type="italics"/>
                  gr.
                    <emph.end type="italics"/>
                  Completa igitur quarta parte revolutio­
                    <lb/>
                  nis unius corpus perveniet ad Apſidem imam, & completa alia
                    <lb/>
                  quarta parte ad Apſidem ſummam, & ſic deinceps per vices in
                    <lb/>
                  infinitum. </s>
                  <s>Id quod etiam ex Propoſitione x. </s>
                  <s>manifeſtum eſt. </s>
                  <s>Nam
                    <lb/>
                  corpus urgente hac vi centripeta revolvetur in Ellipſi immobili,
                    <lb/>
                  cujus centrum eſt in centro virium. </s>
                  <s>Quod ſi vis centripeta ſit reci­
                    <lb/>
                  proce ut diſtantia, id eſt directe ut 1/A ſeu (A
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  /A
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  ), erit
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  æqualis 2, ad­
                    <lb/>
                  eoQ.E.I.ter Apſidem ſummam & imam angulus erit graduum (180/√2)
                    <lb/>
                  ſeu 127
                    <emph type="italics"/>
                  gr.
                    <emph.end type="italics"/>
                  16
                    <emph type="italics"/>
                  m.
                    <emph.end type="italics"/>
                  45
                    <emph type="italics"/>
                  ſec.
                    <emph.end type="italics"/>
                  & propterea corpus tali vi revolvens, perpe­
                    <lb/>
                  tua anguli hujus repetitione, vicibus alternis ab Apſide ſumma ad
                    <lb/>
                  imam & ab ima ad ſummam perveniet in æternum. </s>
                  <s>Porro ſi vis
                    <lb/>
                  centripeta ſit reciproce ut latus quadrato-quadratum undecimæ
                    <lb/>
                  dignitatis altitudinis, id eſt reciproce ut A (11/4), adeoQ.E.D.recte ut
                    <lb/>
                  (1/A
                    <emph type="sup"/>
                  11/4
                    <emph.end type="sup"/>
                  ) ſeu ut (A
                    <emph type="sup"/>
                  1/4
                    <emph.end type="sup"/>
                  /A
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  ) erit
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  æqualis 1/4, & (180/√
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  )
                    <emph type="italics"/>
                  gr.
                    <emph.end type="italics"/>
                  æqualis 360
                    <emph type="italics"/>
                  gr.
                    <emph.end type="italics"/>
                  & prop­
                    <lb/>
                  terea corpus de Apſide ſumma diſcedens & ſubinde perpetuo de­
                    <lb/>
                  ſcendens, perveniet ad Apſidem imam ubi complevit revolutionem
                    <lb/>
                  integram, dein perpetuo aſcenſu complendo aliam revolutionem in­
                    <lb/>
                  regram, redibit ad Apſidem ſummam: & ſic per vices in æternum. </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note104"/>
                  DE MOTU
                    <lb/>
                  CORPORUM</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Exempl.
                    <emph.end type="italics"/>
                  3. Aſſumentes
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  pro quibuſvis indicibus dignitatum
                    <lb/>
                  Altitudinis, &
                    <emph type="italics"/>
                  b, c
                    <emph.end type="italics"/>
                  pro numeris quibuſvis datis, ponamus vim cen­
                    <lb/>
                  tripetam eſſe ut (
                    <emph type="italics"/>
                  b
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                    <emph.end type="sup"/>
                  +
                    <emph type="italics"/>
                  c
                    <emph.end type="italics"/>
                  A
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                    <emph.end type="sup"/>
                  /A
                    <emph type="italics"/>
                  cub.
                    <emph.end type="italics"/>
                  ), id eſt, ut (
                    <emph type="italics"/>
                  b
                    <emph.end type="italics"/>
                  in —T-X
                    <emph type="sup"/>
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                    <emph.end type="sup"/>
                  +
                    <emph type="italics"/>
                  c
                    <emph.end type="italics"/>
                  in —T-X
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                    <emph.end type="sup"/>
                  /A
                    <emph type="italics"/>
                  cub.
                    <emph.end type="italics"/>
                  )
                    <lb/>
                  ſeu (per eandem Methodum noſtram Serierum convergentium) ut
                    <lb/>
                  (
                    <emph type="italics"/>
                  b
                    <emph.end type="italics"/>
                  T
                    <emph type="sup"/>
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                    <emph.end type="sup"/>
                  +
                    <emph type="italics"/>
                  c
                    <emph.end type="italics"/>
                  T
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                    <emph.end type="sup"/>
                  -
                    <emph type="italics"/>
                  mb
                    <emph.end type="italics"/>
                  XT
                    <emph type="sup"/>
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                  -1
                    <emph.end type="sup"/>
                  -
                    <emph type="italics"/>
                  nc
                    <emph.end type="italics"/>
                  XT
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  -1
                    <emph.end type="sup"/>
                  +(
                    <emph type="italics"/>
                  mm-mb
                    <emph.end type="italics"/>
                  /2)XXT
                    <emph type="sup"/>
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                  -2
                    <emph.end type="sup"/>
                  +(
                    <emph type="italics"/>
                  nn-nc
                    <emph.end type="italics"/>
                  /2)XXT
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  -2
                    <emph.end type="sup"/>
                    <emph type="italics"/>
                  &c.
                    <emph.end type="italics"/>
                  /A
                    <emph type="italics"/>
                  cub.
                    <emph.end type="italics"/>
                  ) </s>
                </p>
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