Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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                <p type="main">
                  <s>
                    <pb xlink:href="039/01/265.jpg" pagenum="237"/>
                  recta illa
                    <emph type="italics"/>
                  b, PC a, PQ c, CH e
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  CD o
                    <emph.end type="italics"/>
                  ; rectangulum
                    <emph type="italics"/>
                  a+o
                    <emph.end type="italics"/>
                    <lb/>
                    <arrow.to.target n="note213"/>
                  in
                    <emph type="italics"/>
                  c-a-o
                    <emph.end type="italics"/>
                  ſeu
                    <emph type="italics"/>
                  ac-aa-2ao+co-oo
                    <emph.end type="italics"/>
                  æquale eſt rectangulo
                    <lb/>
                    <emph type="italics"/>
                  b
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  DI,
                    <emph.end type="italics"/>
                  adeoque
                    <emph type="italics"/>
                  DI
                    <emph.end type="italics"/>
                  æquale
                    <emph type="italics"/>
                  (ac-aa/b)+(c-2a/b)o-(oo/b).
                    <emph.end type="italics"/>
                  Jam ſcri­
                    <lb/>
                  bendus eſſet hujus ſeriei ſecundus terminus
                    <emph type="italics"/>
                  (c-2a/b)o
                    <emph.end type="italics"/>
                  pro Q
                    <emph type="italics"/>
                  o,
                    <emph.end type="italics"/>
                  ter­
                    <lb/>
                  tius item terminus (
                    <emph type="italics"/>
                  oo/b
                    <emph.end type="italics"/>
                  ) pro R
                    <emph type="italics"/>
                  oo.
                    <emph.end type="italics"/>
                  Cum vero plures non ſint ter­
                    <lb/>
                  mini, debebit quarti coefficiens S evaneſcere, & propterea quan­
                    <lb/>
                  titas (S/R√1+QQ) cui Medii denſitas proportionalis eſt, nihil
                    <lb/>
                  erit. </s>
                  <s>Nulla igitur Medii denſitate movebitur Projectile in Para­
                    <lb/>
                  bola, uti olim demonſtravit
                    <emph type="italics"/>
                  Galilæus, Q.E.I.
                    <emph.end type="italics"/>
                  </s>
                </p>
                <p type="margin">
                  <s>
                    <margin.target id="note213"/>
                  LIBER
                    <lb/>
                  SECUNDUS.</s>
                </p>
                <p type="main">
                  <s>
                    <emph type="italics"/>
                  Exempl.
                    <emph.end type="italics"/>
                  3. Sit linea
                    <emph type="italics"/>
                  AGK
                    <emph.end type="italics"/>
                  Hyperbola, Aſymptoton habens
                    <lb/>
                    <emph type="italics"/>
                  NX
                    <emph.end type="italics"/>
                  plano horizontali
                    <emph type="italics"/>
                  AK
                    <emph.end type="italics"/>
                  perpendicularem; & quæratur Medii
                    <lb/>
                  denſitas quæ faciat ut Projectile moveatur in hac linea. </s>
                </p>
                <p type="main">
                  <s>Sit
                    <emph type="italics"/>
                  MX
                    <emph.end type="italics"/>
                  Aſymptotos altera, ordinatim applicatæ
                    <emph type="italics"/>
                  DG
                    <emph.end type="italics"/>
                  productæ
                    <lb/>
                  occurrens in
                    <emph type="italics"/>
                  V,
                    <emph.end type="italics"/>
                  & ex natura Hyperbolæ, rectangulum
                    <emph type="italics"/>
                  XV
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  VG
                    <emph.end type="italics"/>
                    <lb/>
                  dabitur. </s>
                  <s>Datur autem ratio
                    <emph type="italics"/>
                  DN
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  VX,
                    <emph.end type="italics"/>
                  & propterea datur etiam
                    <lb/>
                  rectangulum
                    <emph type="italics"/>
                  DN
                    <emph.end type="italics"/>
                  in
                    <emph type="italics"/>
                  VG.
                    <emph.end type="italics"/>
                  Sit illud
                    <emph type="italics"/>
                  bb
                    <emph.end type="italics"/>
                  ; & completo parallelogrammo
                    <lb/>
                    <emph type="italics"/>
                  DNXZ,
                    <emph.end type="italics"/>
                  dicatur
                    <emph type="italics"/>
                  BN a, BD o, NX c,
                    <emph.end type="italics"/>
                  & ratio data
                    <emph type="italics"/>
                  VZ
                    <emph.end type="italics"/>
                  ad
                    <emph type="italics"/>
                  ZX
                    <emph.end type="italics"/>
                    <lb/>
                  vel
                    <emph type="italics"/>
                  DN
                    <emph.end type="italics"/>
                  ponatur eſſe
                    <emph type="italics"/>
                  m/n
                    <emph.end type="italics"/>
                  . </s>
                  <s>Et erit
                    <emph type="italics"/>
                  DN
                    <emph.end type="italics"/>
                  æqualis
                    <emph type="italics"/>
                  a-o, VG
                    <emph.end type="italics"/>
                  æqualis
                    <lb/>
                    <emph type="italics"/>
                  (bb/a-o), VZ
                    <emph.end type="italics"/>
                  æqualis
                    <emph type="italics"/>
                  m/n—a-o,
                    <emph.end type="italics"/>
                  &
                    <emph type="italics"/>
                  GD
                    <emph.end type="italics"/>
                  ſeu
                    <emph type="italics"/>
                  NX-VZ-VG
                    <emph.end type="italics"/>
                  æ­
                    <lb/>
                  qualis
                    <emph type="italics"/>
                  c-m/n a+m/n o-(bb/a-o).
                    <emph.end type="italics"/>
                  Reſolvatur terminus (
                    <emph type="italics"/>
                  bb/a-o
                    <emph.end type="italics"/>
                  ) in ſeriem
                    <lb/>
                  convergentem
                    <emph type="italics"/>
                  (bb/a)+(bb/aa)o+(bb/a
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  )oo+(bb/a
                    <emph type="sup"/>
                  4
                    <emph.end type="sup"/>
                  )o
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>& ſiet
                    <emph type="italics"/>
                  GD
                    <emph.end type="italics"/>
                  æqua­
                    <lb/>
                  lis
                    <emph type="italics"/>
                  c-m/n a-(bb/a)+m/n o-(bb/aa)o-(bb/a
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  )o
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  -(bb/a
                    <emph type="sup"/>
                  4
                    <emph.end type="sup"/>
                  )o
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                    <emph.end type="italics"/>
                  &c. </s>
                  <s>Hujus ſeriei termi­
                    <lb/>
                  nus ſecundus
                    <emph type="italics"/>
                  m/no-(bb/aa)o
                    <emph.end type="italics"/>
                  uſurpandus eſt pro Q
                    <emph type="italics"/>
                  o,
                    <emph.end type="italics"/>
                  tertius cum ſigno
                    <lb/>
                  mutato
                    <emph type="italics"/>
                  (bb/a
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  )o
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                    <emph.end type="italics"/>
                  pro R
                    <emph type="italics"/>
                  o
                    <emph.end type="italics"/>
                    <emph type="sup"/>
                  2
                    <emph.end type="sup"/>
                  , & quartus cum ſigno etiam mutato
                    <emph type="italics"/>
                  (bb/a
                    <emph type="sup"/>
                  4
                    <emph.end type="sup"/>
                  )o
                    <emph type="sup"/>
                  1
                    <emph.end type="sup"/>
                    <emph.end type="italics"/>
                    <lb/>
                  pro S
                    <emph type="italics"/>
                  o
                    <emph.end type="italics"/>
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  , eorumque coefficientes
                    <emph type="italics"/>
                  m/n-(bb/aa), (bb/a
                    <emph type="sup"/>
                  3
                    <emph.end type="sup"/>
                  )
                    <emph.end type="italics"/>
                  & (
                    <emph type="italics"/>
                  bb/a
                    <emph type="sup"/>
                  4
                    <emph.end type="sup"/>
                    <emph.end type="italics"/>
                  ) ſcribendæ ſunt
                    <lb/>
                  in Regula ſuperiore, pro Q, R & S. </s>
                  <s>Quo facto prodit medii denſitas </s>
                </p>
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