Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                  <s>
                    <pb xlink:href="039/01/157.jpg" pagenum="129"/>
                  & collatis numeratorum terminis, fiet RGG-RFF+TFF
                    <lb/>
                    <arrow.to.target n="note105"/>
                  ad
                    <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"/>
                  , ut -FF ad -
                    <emph type="italics"/>
                  mb
                    <emph.end type="italics"/>
                  T
                    <emph type="sup"/>
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                  -1
                    <emph.end type="sup"/>
                  -
                    <emph type="italics"/>
                  nc
                    <emph.end type="italics"/>
                  T
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  -1
                    <emph.end type="sup"/>
                    <lb/>
                  +(
                    <emph type="italics"/>
                  mm-m
                    <emph.end type="italics"/>
                  /2)
                    <emph type="italics"/>
                  b
                    <emph.end type="italics"/>
                  XT
                    <emph type="sup"/>
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                  -2
                    <emph.end type="sup"/>
                  +(
                    <emph type="italics"/>
                  nn-n
                    <emph.end type="italics"/>
                  /2)
                    <emph type="italics"/>
                  c
                    <emph.end type="italics"/>
                  XT
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  -2
                    <emph.end type="sup"/>
                  &c. </s>
                  <s>Et ſumendo rationes ulti­
                    <lb/>
                  mas quæ prodeunt ubi Orbes ad formam circularem accedunt, fit
                    <lb/>
                  GG ad
                    <emph type="italics"/>
                  b
                    <emph.end type="italics"/>
                  T
                    <emph type="sup"/>
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                  -1
                    <emph.end type="sup"/>
                  +
                    <emph type="italics"/>
                  c
                    <emph.end type="italics"/>
                  T
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  -1
                    <emph.end type="sup"/>
                  , ut FF ad
                    <emph type="italics"/>
                  mb
                    <emph.end type="italics"/>
                  T
                    <emph type="sup"/>
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                  -1
                    <emph.end type="sup"/>
                  +
                    <emph type="italics"/>
                  nc
                    <emph.end type="italics"/>
                  T
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  -1
                    <emph.end type="sup"/>
                  , &
                    <lb/>
                  viciſſim GG ad FF ut
                    <emph type="italics"/>
                  b
                    <emph.end type="italics"/>
                  T
                    <emph type="sup"/>
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                  -1
                    <emph.end type="sup"/>
                  +
                    <emph type="italics"/>
                  c
                    <emph.end type="italics"/>
                  T
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  -1
                    <emph.end type="sup"/>
                  ad
                    <emph type="italics"/>
                  mb
                    <emph.end type="italics"/>
                  T
                    <emph type="sup"/>
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                  -1
                    <emph.end type="sup"/>
                  +
                    <emph type="italics"/>
                  nc
                    <emph.end type="italics"/>
                  T
                    <emph type="sup"/>
                    <emph type="italics"/>
                  n
                    <emph.end type="italics"/>
                  -1
                    <emph.end type="sup"/>
                  . </s>
                  <s>
                    <lb/>
                  Quæ proportio, exponendo altitudinem maximam
                    <emph type="italics"/>
                  CV
                    <emph.end type="italics"/>
                  ſeu T Arith­
                    <lb/>
                  metice per Unitatem, fit GG ad FF ut
                    <emph type="italics"/>
                  b+c
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  mb+nc,
                    <emph.end type="italics"/>
                  adeoque ut
                    <lb/>
                  1 ad (
                    <emph type="italics"/>
                  mb+nc/b+c
                    <emph.end type="italics"/>
                  ). Unde eſt G ad F, id eſt angulus
                    <emph type="italics"/>
                  VCp
                    <emph.end type="italics"/>
                  ad angulum
                    <lb/>
                    <emph type="italics"/>
                  VCP,
                    <emph.end type="italics"/>
                  ut 1 ad √(
                    <emph type="italics"/>
                  mb+nc/b+c
                    <emph.end type="italics"/>
                  ). Et propterea cum angulus
                    <emph type="italics"/>
                  VCP
                    <emph.end type="italics"/>
                  inter
                    <lb/>
                  Apſidem ſummam & Apſidem imam in Ellipſi immobili ſit 180
                    <emph type="italics"/>
                  gr.
                    <emph.end type="italics"/>
                    <lb/>
                  erit angulus
                    <emph type="italics"/>
                  VCp
                    <emph.end type="italics"/>
                  inter eaſdem Apſides, in Orbe quem corpus vi
                    <lb/>
                  centripeta quantitati (
                    <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"/>
                  ) proportionali deſcribit, æqua­
                    <lb/>
                  lis angulo graduum 180 √(
                    <emph type="italics"/>
                  b+c/mb+nc
                    <emph.end type="italics"/>
                  ). Et eodem argumento ſi vis cen­
                    <lb/>
                  tripeta ſit 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"/>
                  ), angulus inter Apſides invenietur graduum
                    <lb/>
                  180 √(
                    <emph type="italics"/>
                  b-c/mb-nc
                    <emph.end type="italics"/>
                  ). Nec ſecus reſolvetur Problema in caſibus diffi­
                    <lb/>
                  cilioribus. </s>
                  <s>Quantitas cui vis centripeta proportionalis eſt, re­
                    <lb/>
                  ſolvi ſemper debet in Series convergentes denominatorem ha­
                    <lb/>
                  bentes A
                    <emph type="italics"/>
                  cub.
                    <emph.end type="italics"/>
                  Dein pars data numeratoris qui ex illa operatione
                    <lb/>
                  provenit ad ipſius partem alteram non datam, & pars data nu­
                    <lb/>
                  meratoris hujus RGG-RFF+TFF-FFX ad ipſius partem
                    <lb/>
                  alteram non datam in eadem ratione ponendæ ſunt: Et quantitates
                    <lb/>
                  ſuperfluas delendo, ſcribendoque Unitatem pro T, obtinebitur
                    <lb/>
                  proportio G ad F. </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note105"/>
                  LIBER
                    <lb/>
                  PRIMUS.</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Corol.
                    <emph.end type="italics"/>
                  1. Hinc ſi vis centripeta ſit ut aliqua altitudinis digNI­
                    <lb/>
                  tas, inveniri poteſt dignitas illa ex motu Apſidum; & contra. </s>
                  <s>
                    <lb/>
                  Nimirum ſi motus totus angularis, quo corpus redit ad Apſidem
                    <lb/>
                  eandem, ſit ad motum angularem revolutionis unius, ſeu graduum
                    <lb/>
                  360, ut numerus aliquis
                    <emph type="italics"/>
                  m
                    <emph.end type="italics"/>
                  ad numerum alium
                    <emph type="italics"/>
                  n,
                    <emph.end type="italics"/>
                  & altitudo no­
                    <lb/>
                  minetur A: erit vis ut altitudinis dignitas illa A
                    <emph type="sup"/>
                  (
                    <emph type="italics"/>
                  nn/mm
                    <emph.end type="italics"/>
                  )-3
                    <emph.end type="sup"/>
                  , cujus In-</s>
                </p>
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